Locally convex quasi star-algebras and their representations 9783030377045, 9783030377052, 2352372372

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Lecture Notes in Mathematics  2257

Maria Fragoulopoulou Camillo Trapani

Locally Convex Quasi *-Algebras and their Representations

Lecture Notes in Mathematics Editors-in-Chief: Jean-Michel Morel, Cachan Bernard Teissier, Paris Advisory Editors: Karin Baur, Leeds Michel Brion, Grenoble Camillo De Lellis, Princeton Alessio Figalli, Zurich Annette Huber, Freiburg Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Cambridge Angela Kunoth, Cologne Ariane Mézard, Paris Mark Podolskij, Aarhus Sylvia Serfaty, New York Gabriele Vezzosi, Florence Anna Wienhard, Heidelberg

2257

More information about this series at http://www.springer.com/series/304

Maria Fragoulopoulou • Camillo Trapani

Locally Convex Quasi *-Algebras and their Representations

Maria Fragoulopoulou Department of Mathematics National and Kapodistrian University of Athens Athens, Greece

Camillo Trapani Matematica e Informatica Universit`a degli Studi di Palermo Palermo, Italy

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-030-37704-5 ISBN 978-3-030-37705-2 (eBook) https://doi.org/10.1007/978-3-030-37705-2 Mathematics Subject Classification (2010): Primary: 46H05, 46H15, 46H20; Secondary: 47L60 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Algebraic Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Quasi *-Algebras and Partial *-Algebras. . . . . . . . . . . . . . . . . . . . . 2.1.2 Basic Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Quasi *-Algebras and Partial *-Algebras of Operators . . . . . . 2.2 Homomorphisms, Ideals and Representations . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Families of Sesquilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Construction of *-Representations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 GNS-Like Construction with Sesquilinear Forms . . . . . . . . . . . 2.4.2 GNS-Like Construction with Linear Functionals . . . . . . . . . . . .

9 9 9 11 13 14 17 19 20 22

3

Normed Quasi *-Algebras: Basic Theory and Examples . . . . . . . . . . . . . . . . 3.1 Basic Definitions and Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Auxiliary Seminorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Sufficient Family of Forms and *-Semisimplicity . . . . . . . . . . . 3.2 Continuity of Representable Linear Functionals . . . . . . . . . . . . . . . . . . . . . . 3.3 Continuity of *-Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 33 36 40 54 60

4

Normed Quasi *-Algebras: Bounded Elements and Spectrum . . . . . . . . . 4.1 The *-Algebra of Bounded Elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Bounded Elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Normal Banach Quasi *-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Strongly Bounded Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Spectrum of an Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 The Inverse of an Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 The Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 The *-Semisimple Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 67 67 70 75 80 80 83 85

v

vi

5

Contents

CQ*-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Basic Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Commutative CQ*-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 General CQ*-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Construction of CQ*-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Constructive Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Starting from a *-Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Construction of CQ*-Algebras through Families of Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 *-Homomorphisms of CQ*-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 CQ*-Algebras and Left Hilbert Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 The Structure of Strict CQ*-Algebras . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Measurable Operators and CQ*-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Noncommutative Measure and Integration . . . . . . . . . . . . . . . . . . . 5.6.2 Noncommutative Lp -Spaces as Proper CQ*-Algebras . . . . . . 5.6.3 CQ*-Algebras over Finite von Neumann Algebras . . . . . . . . . . 5.6.4 A First Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 95 96 102 108 108 111 113 116 121 124 126 126 128 130 135

6

Locally Convex Quasi *-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Representable Functionals on Locally Convex Quasi *-Algebras . . . 6.2 Order Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Fully Representable Quasi *-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Ordered Bounded Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Partial Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 143 147 150 154 158

7

Locally Convex Quasi C*-Algebras and Their Structure . . . . . . . . . . . . . . . 7.1 Locally Convex Quasi C*-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Locally Convex Quasi C*-Algebras of Operators . . . . . . . . . . . . . . . . . . . . 7.3 Structure of Commutative Locally Convex Quasi C*-Algebras . . . . . 7.4 Functional Calculus for Positive Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Structure of Noncommutative Locally Convex Quasi C*-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Locally Convex Quasi C*-Algebras and Noncommutative Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 The Representation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 163 170 175 184

A *-Algebras and Representations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Algebras: Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.1 Hermitian and Positive Linear Functionals. . . . . . . . . . . . . . . . . . . A.2.2 The Gelfand–Naimark–Segal Theorem . . . . . . . . . . . . . . . . . . . . . . A.2.3 Boundedness of the GNS Representation; Admissibility . . . . A.3 Spectral Radius, Spectrum and All That . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Admissible Positive Linear Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 C*-Seminorms on Banach *-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Gelfand Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201 201 204 209 211 215 218 225 228 230

189 194 197

Contents

vii

A.6.1 A.6.2

The Commutative Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 The Noncommutative Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

B Operators in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Bounded Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 The Adjoint of a Bounded Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3.1 Symmetric Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3.2 Projection Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3.3 Isometric and Unitary Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4 Unbounded Operators in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.5 Symmetric and Selfadjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

237 237 240 242 242 243 243 243 247

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

Chapter 1

Introduction

A locally convex quasi *-algebra is a pair consisting of a locally convex space A[τ ] containing densely a *-algebra A0 , whose multiplication and involution extend to A, in the sense that for any a ∈ A and x ∈ A0 , the elements ax, xa belong to A, in such a way that (ax)∗ = x ∗ a ∗ , (xa)∗ = a ∗ x ∗ and moreover the left and right multiplications of the elements of A by a fixed x ∈ A0 , as well as the involution on A[τ ] are continuous. A typical example is obtained by taking as A the completion of a locally convex *-algebra A0 [τ ] with continuous involution and separately continuous multiplication. The book in hands aims to present the essential aspects of the theory of locally convex quasi *-algebras as it has been developed so far. We begin with giving a brief account of their history and of the motivations behind their initiation. Even though there are several familiar instances, where the preceding structure appears in a natural way (consider, for instance, the Lp spaces on an interval or the spaces of distributions), until the 1980s only a little attention had been paid, in general, to the interplay between the ‘partially defined multiplication’ and the topological structure they possess. The theory of Banach and C*-algebras was recent enough (Banach algebras appeared in 1938 with Gelfand’s thesis and C*-algebras first appeared in 1943 in the famous paper of Gelfand and Naimark [60]) to let researchers take into account a structure with a partial multiplication, looking as an exotic feature with very few or not so exciting consequences. Actually, the beauty of the theory of C*-algebras consists in the fact that an apparently simple condition, the wellknown C*-condition, has such a large number of nontrivial consequences. The noncommutative Gelfand–Naimark theorem crowns the theory by showing that any C*-algebra can be realized as a C*-algebra of bounded linear operators on a Hilbert space. The main outcome of this is a unification of an abstract set-up with the theory of operator algebras, mostly developed by von Neumann. This deep result made of C*-algebras a very popular research subject evolving in several different and sometimes unexpected directions. Today, when thousands of papers on this subject © Springer Nature Switzerland AG 2020 M. Fragoulopoulou, C. Trapani, Locally Convex Quasi *-Algebras and their Representations, Lecture Notes in Mathematics 2257, https://doi.org/10.1007/978-3-030-37705-2_1

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1 Introduction

are at our disposal, we have a strong awareness of the wide range of applications that C*-algebras have in many fields of mathematics (noncommutative geometry, noncommutative analysis, harmonic analysis, group representations) and in physics (quantum statistical mechanics, quantum field theory). The first suggestions to go towards partial algebraic structures came from quantum physics and from the theory of representations of Lie algebras. However, apart from the short discussion below, the reader will not find any reference to these theories in this book. Nevertheless, many books and papers on these subjects are listed in the bibliography. But, let us now come back to our brief historic overview. In 1964 Haag and Kastler [61] proposed the so-called algebraic approach to quantum theories, whose cornerstone is the assumption that the local observables of a quantum system (that is physical quantities that can be measured within a region of finite measure) constitute a C*-algebra and global quantities are intended to be limits in the C*-norm of nets of local observables. Representing the C*algebra of local observables with (necessarily) bounded operators, this approach provides the recipe for theoretical physicists to develop their computations on physical systems. So, after the publication of Haag and Kastler’s paper, the research of physical models fitting into their set-up, has been on the stage and for a large number of models, mostly taken from quantum field theory and quantum statistical mechanics, this attempt has been successful. On the other hand, several models, like the Bose gas [40], go beyond this algebraic framework. This happens, in general, for models, where unbounded operators play a relevant role, and the representation of observables by means of bounded operators determines, in a sense, loss of information. One possible way to include a larger number of models in the formulation of Haag and Kastler is to enlarge the algebraic set-up, considering also families of unbounded operators (and not only von Neumann algebras) for the representation of local observable algebras. For these reasons, algebras of unbounded operators, the so-called O*-algebras (see [42, 65, 72, 91] and [23], for a complete overview), partial *-algebras [2, 36] and quasi *-algebras [67, 68] have been studied, mainly from the mathematical point of view, deeply enough as to achieve the status of complete theories. In particular, the appearance of partial *-algebras and quasi *algebras found its impetus in the non-neglectable number of instances in quantum theories, where one has to deal with sets of operators, whose multiplication is not defined for arbitrary pairs of elements. This unpleasant feature depends essentially on two facts: first, the operators, being in general, unbounded cannot be defined on the whole Hilbert space of states; second, they may not have a common invariant dense domain of definition (the existence of local von Neumann algebras, which do not leave the natural domain of Wightman fields invariant, has been shown by Horuzhy and Voronin in [62]). Following an idea of Borchers [56], Antoine and Karwowski [36] began an analysis of partial *-algebras, which was then continued in a long series of papers with Inoue, Mathot and one of us (CT) (see, e.g., [34, 35, 37, 38]). In 2002 a monograph [2] on this subject appeared, where the results obtained so far were systematically collected.

1 Introduction

3

The problem of a rigorous mathematical description of the thermodynamical limit of the local Heisenberg dynamics for certain quantum statistical systems led Lassner, more or less in the same period, to introduce the notion of a quasi *-algebra [67, 68], which from the algebraic point of view, is a structure simpler than that of a general partial *-algebra and can be more easily cast into a topological framework. As mentioned before, the most basic example of a locally convex quasi *-algebra is the completion of a locally convex *-algebra, whose involution is continuous and multiplication is separately continuous. Completions of this sort may really occur in quantum statistics. In fact, in this case, the observable algebra A, which is supposed to be a C*-algebra, does not contain, in general, the thermodynamical limit of the local Heisenberg dynamics. Then, the procedure to circumvent this difficulty, is to define on A a locally convex topology τ , (which, in general, depends on the model under consideration) in such a way that the dynamics at the thermodynamical limit belongs to the completion of A under τ . More general structures than C*-algebras are also needed if one wants to represent Lie algebras by operators in infinite dimensional Hilbert spaces. In fact, the Heisenberg Lie algebra h generated by three elements a, b, c ∈ h, whose Lie brackets are defined by [a, b] = c,

[a, c] = [b, c] = 0,

does not possess any bounded representation. Actually, the Wiener–Wielandt theorem states that there exists no bounded representation π (with carrier space a Hilbert space H) of the Heisenberg Lie algebra h, satisfying the equality π([a, b]) = I , where I is the identity operator on H. A familiar representation of h is provided by a pair of operators P , Q satisfying the canonical commutation relation (CCR) P Q − QP = −iI , which in physics expresses the Heisenberg uncertainty principle. This relation is nothing, but a special representation π of pairs of operators satisfying the CCR condition, that allow the construction of a locally convex quasi *-algebra of pseudo differential operators (whose coefficients are distributions!) [69]. Operators acting on distribution spaces provide another interesting field of applications of locally convex quasi *-algebras, which is fairly important also from the mathematical point of view. Indeed, these arise in a natural way, when one considers the space L(D, D× ) of continuous operators acting on rigged Hilbert spaces D ⊂ H ⊂ D (under certain assumptions on the topology of D). The most frequent situation in the applications occurs, when D is the space of C ∞ vectors of a self-adjoint operator H . The quasi *-algebra L(D, D× ) plays also a role in Wightman quantum field theory: point-like fields satisfying a certain regularity condition are indeed elements of L(D, D× ), for an appropriate domain D. On the other hand, a smeared field satisfying the usual Wightman axioms on D, gives rise to a point-like field with values in L(D, D× ) [2, Chapter 11], [58]. These considerations motivate a study of locally convex quasi *-algebras, to which this book is devoted. Even though many aspects of the theory are yet unexplored, the amount of results that can be found in the literature is quite large

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1 Introduction

and in authors’ opinion it is time for a monographic synthesis on the subject, so that any interested researchers could have a reference guide to the theory developed so far. This book is, in a sense, a cadet child of the monograph [2] coauthored by one of us (CT) and appeared in 2002. Like every child, it inherited something from his parents, but not so much as to be indistinguishable and, after all, seventeen years have not passed in vain. The book [2] was mainly concerned, as the title itself announced, with representations by means of families of unbounded operators. Only in Chap. 7 locally convex topologies on partial *-algebras were considered. Here, on the contrary the aspects of representation theory, which do not need topology, cover only Chap. 2. We do not claim originality of the contents of this book: they have mostly appeared in journal articles, with the possible exception of some parts of spectral theory, which are published here first time (a subsection in Chap. 4). Chapter 2 is entirely dedicated to the algebraic aspects of the theory. After giving the basic definitions and properties of quasi *-algebras, we present a study of their *-representations. A *-representation of a given quasi *-algebra (A, A0 ) is a *-homomorphism π from (A, A0 ) into the partial *-algebra L† (D, H) of closable linear operators defined on a dense domain D of a Hilbert space H. These representations may be unbounded, in the sense that for some a ∈ A, π(a) can be a true unbounded operator in H. A standard way for studying *-representations is the use of the Gelfand–Naimark–Segal (GNS) construction, which allows to retrieve *-representations from certain positive linear functionals. The lack of an everywhere defined multiplication makes the situation more convenient, by replacing linear functionals with a family of positive sesquilinear forms on A satisfying certain invariance properties. The possibility of performing a Hilbert space *-representation, starting either from an element of this class of positive sesquilinear forms, or from certain positive linear functionals on A, called representable, the classical GNS procedure is discussed, without having to assume, a priori, the existence of a topology on A. In Chap. 3 we begin the analysis of locally convex quasi *-algebras starting from the simplest case: normed and Banach quasi *-algebras. This situation covers very familiar examples such as Lp -spaces (both commutative and noncommutative). In this chapter, we will mainly consider the case, where (A[·], A0 ) is a Banach quasi *-algebra. This means, in a few words, that A is a Banach space, whose norm  ·  satisfies certain coupling properties related to the partial multiplication of (A[ · ], A0 ). The main goal is the study of structure of normed quasi *-algebras and particular attention is paid to the role of certain families of positive sesquilinear forms and to the *-representations they define. Several examples are also discussed. In particular, since the range L† (D, H) of *-representations carries a number of topologies, making of it a locally convex partial *-algebra, it makes sense to consider the problem of continuity of *-representations on topological quasi *-algebras. It turns out that in the case of a normed quasi *-algebra, this problem is closely linked with the normed structure of the quasi *-algebra under consideration: thus, as in the case of *-representations of Banach *-algebras, a certain amount of information on

1 Introduction

5

the structure of a Banach quasi *-algebra can be obtained from the knowledge of the properties of the family of its *-representations. Chapter 4 is devoted to the special role the set Ab of bounded elements of a normed quasi *-algebra (A[·], A0 ) plays in the study of its structure. Bounded elements are characterized by the fact that the corresponding multiplication operators are bounded linear maps. More precisely, an element a ∈ A is said to be bounded if both La : x ∈ A0 [ · ] → ax ∈ A[ · ] and Ra : x ∈ A0 [ · ] → xa ∈ A[ · ] are bounded linear maps. Then, we focus our attention to the class of normal Banach quasi *-algebras: they are characterized by the fact that Ab is a Banach *-algebra. If (A[·], A0 ) is normal, the Banach *-algebra Ab allows the notion of spectrum of an element a ∈ A, which enjoys properties analogous to the spectrum of an element in a Banach *-algebra. An important role in this study, in particular for the analysis of *-representations, is played by two seminorms p, q (defined in Chap. 3) that emulate the Gelfand–Naimark seminorm on a Banach *-algebra (but q is only defined on a domain D(q) ⊆ A: it is actually an unbounded C*-seminorm, in the sense of [46, 55, 63, 78]). This approach, already extensively used in the study of locally convex *-algebras [54, 55, 59] has given a quite deep insight into their structure, in particular for the existence of well-behaved *-representations (see, also [54, 74]). Chapter 5 deals with CQ*-algebras. From the historical point of view, the first results on the Banach case were obtained by F. Bagarello and CT, in 1990s [48, 49], for the so called CQ*-algebras, which roughly speaking, are completions of C*algebras with respect to a second norm defining a topology coarser than that induced by the C*-norm. At the beginning, a quite general construction was performed, giving large room to the case, where the involution of the C*-algebra was not necessarily the same like that of the Banach space enclosing the whole structure. This may appear to be a little artificial, but in a cooperation of F. Bagarello, A. Inoue and CT was realized that this was the right environment, where some aspects of the Tomita–Takesaki theory could be naturally cast [47]. On the other hand, the family of proper CQ*-algebras enters more directly in the framework of locally convex quasi *-algebras and a lot of results were demonstrated. Here, it is worth mentioning the characterization of *-semisimple proper CQ*-algebras in terms of function spaces [50] in the commutative case, or in terms of measurable operators in the noncommutative case [52]. With Chap. 6, we finally go to a more general set-up. The main topic is, in fact, to give conditions under which a locally convex quasi *-algebra (A[τ ], A0 ) attains sufficiently many (τ, tw )-continuous ∗ -representations in L† (D, H), to separate its points. This leads to the notion of fully-representable quasi *-algebras, which are explored in detail. Once we have at our disposal a sufficiently large family of *representations, a usual notion of bounded elements on A[τ ] rises. Of course, this generalizes what was done in Chap. 5, for the Banach case. On the other hand, a natural order exists on (A[τ ], A0 ) related to the topology τ , which also leads to a different kind of bounded elements, which we call order bounded. Under certain conditions, the latter notion of boundedness coincides with the usual one. In a fully representable quasi *-algebra (A[τ ], A0 ) a weak partial multiplication  can be introduced and an unbounded C*-seminorm  · b is defined on A (by means of

6

1 Introduction

the order boundedness), with domain the partial *-subalgebra Ab of A[τ ] consisting of all order bounded elements. In this way, under certain conditions, Ab becomes a C*-algebra, with respect to the weak multiplication  and the C*-norm  · b . Chapter 7 is closely connected with Chaps. 5 and 6. In fact, the attention is again focused on the completion of a given C*-algebra A0 , but this time with respect to a second topology that is not necessarily a normed one. Clearly, the techniques to be used become more and more complicated and also the understanding becomes more difficult. The study of the structure and representation theory of the completion of a (normed) C*-algebra A0 [ · 0 ], under a locally convex *-algebra topology τ on A0 , was started in [44] and continued in [59]. When the multiplication of A0 , 0 [τ ] of A0 [τ ] is a GB*with respect to τ is jointly continuous, the completion A algebra (of unbounded operators) over the unit ball U (A0 ) = {x ∈ A0 : x0 ≤ 1} of A0 [ · 0 ], if and only if, U (A0 ) is τ -closed [59, Corollary 2.2]. When the 0 [τ ] may multiplication of A0 , with respect to τ is just separately continuous, A fail to be a locally convex ∗-algebra, but may well carry the structure of a quasi 0 [τ ], ∗-algebra. First studies on the properties and the ∗-representation theory of A in this case, were done in [44, Section 3] and [59, Section 3]. These studies have led to the introduction of locally convex quasi C*-algebras, which are discussed in this chapter. As usually, a more general set-up requires a strengthening of several notions and adaptation of many others. This is the reason why two notions of positivity 0 [τ ]. One is called just are introduced in the locally convex quasi *-algebra A “positivity” and the second one “commutatively positivity” (see Definition 7.1.1 and after that discussion before (T3 )). Locally convex quasi C*-algebras are then defined and some examples from various classes of topological algebras are discussed. The chapter continues with a study of locally convex quasi C*-algebras of operators and with the analysis of the structure of commutative locally convex quasi C*-algebras, which is discussed by taking into account [32, Section 6] and [44, 59]. By applying the results of the previous sections and also some ideas developed in [57, Section 4] and [59] one obtains a functional calculus for the positive elements of a commutative locally convex quasi C*-algebra. Furthermore, if A[τ ] is a noncommutative locally convex quasi C*-algebra, necessary and sufficient conditions are given for A[τ ] to be continuously embedded in a locally convex quasi C*-algebra of operators, obtaining thus Gelfand–Naimark type theorems for the aforementioned topological quasi *-algebras. In the final part of this chapter, some results concerning the possibility of representing a noncommutative locally convex quasi C*-algebra as locally convex quasi *-algebra of operators, measurable in the sense of Segal and Nelson, are presented. Two appendices close this book: in the first one we collect results on *algebras and their representations, so that the reader can find a handy reference to classical results of the theory of Banach (*-)algebras, as well as of C*-algebras that are systematically used throughout this volume. The second appendix is a short exhibition of the basic theorems and facts of (bounded and unbounded) operator theory. We believe they can be useful, not only for a better comprehension of this book, but also as a guide text for mini courses on these topics. Unfortunately the contents of the two appendices do not exhaust what is needed for reading this book:

1 Introduction

7

familiarity is also presumed with general functional analysis, harmonic analysis, operator theory and von Neumann algebras theory. Finally, at the end of the book, we add a list of further references, under the name “Suggested Readings”, in order to provide the young researchers with an adequate literature that will possibly help them to explore further, topological (quasi) *-algebras and their applications. We also remark that the bibliography of [2] provides the reader with an extensive literature covering the representation theory of unbounded operator algebras and not only. Last but not least, we want to extend our heartfelt thanks to all those, who have cooperated with us in these years in writing the papers from which this book originated and also to colleagues and students involved with us into discussions on the topics of the present work. In the end, we want to express our gratitude to the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni of the Istituto Nazionale di Alta Matematica, whose financial support has allowed us several mutual visits during the preparation of this book.

Chapter 2

Algebraic Aspects

The notion of a quasi *-algebra (see Definition 2.1.1 below) was first introduced by G. Lassner in the early 80s of last century ([67, 68], see also [23]). In this chapter, we discuss some algebraic aspects of the theory of these *-algebras. All vector spaces or algebras considered throughout this book are over the field C of complexes and all locally convex spaces are supposed to be Hausdorff. Our basic definitions and notation mainly come from [2].

2.1 Definitions and Examples 2.1.1 Quasi *-Algebras and Partial *-Algebras Definition 2.1.1 A quasi *-algebra (A, A0 ) is a pair consisting of a vector space A and a *-algebra A0 contained in A as a subspace, such that (i) A carries an involution a → a ∗ extending the involution of A0 ; (ii) A is a bimodule over A0 and the module multiplications extend the multiplication of A0 ; In particular, the following associative laws hold: (xa)y = x(ay); a(xy) = (ax)y,

∀ a ∈ A, x, y ∈ A0 ;

(2.1.1)

(iii) (ax)∗ = x ∗ a ∗ , for every a ∈ A and x ∈ A0 . We say that a quasi *-algebra (A, A0 ) is unital, if there is an element e ∈ A0 , such that ae = a = ea, for all a ∈ A; e is unique and called unit of (A, A0 ). We say that (A, A0 ) has a quasi-unit if there exists an element q ∈ A, such that qx = xq = x, for every x ∈ A0 . It is clear that the unit e, if any, is a quasi-unit but the converse is false, in general. © Springer Nature Switzerland AG 2020 M. Fragoulopoulou, C. Trapani, Locally Convex Quasi *-Algebras and their Representations, Lecture Notes in Mathematics 2257, https://doi.org/10.1007/978-3-030-37705-2_2

9

10

2 Algebraic Aspects

A quasi *-algebra is a particular type of a partial *-algebra, in the sense of the following definition. Definition 2.1.2 A partial *-algebra is a complex vector space A, endowed with an involution a → a ∗ (that is, a bijection, such that a ∗∗ = a, for all a ∈ A) and a partial multiplication defined by a set  ⊂ A × A (a binary relation), with the following properties (i) (a, b) ∈  implies (b∗ , a ∗ ) ∈ ; (ii) (a, b1 ), (a, b2 ) ∈  implies (a, λb1 + μb2 ) ∈ , ∀ λ, μ ∈ C; (iii) for any (a, b) ∈ , a product a · b ∈ A is defined, which is distributive with respect to the addition and satisfies the relation (a · b)∗ = b∗ · a ∗ . We shall assume that the partial *-algebra A contains a unit e, if e∗ = e, (e, a) ∈ , ∀ a ∈ A and e · a = a · e = a,

∀ a ∈ A.

If A has no unit, it may always be embedded into a larger partial *-algebra with unit (the so-called unitization of A), in the standard fashion [37]. Given the defining set , spaces of multipliers are defined in obvious way. More precisely, (a, b) ∈  ⇐⇒ a ∈ L(b) i.e., a is a left multiplier of b and b ∈ R(a) i.e., b is a right multiplier of a. By Definition 2.1.2, (ii), L(a), L(b) are clearly vector subspaces of A. For any subset N ⊂ A, we put LN =



L(a),

a∈N

RN =



R(a)

a∈N

and, of course, the involution exchanges ‘L and R’, as follows (LN)∗ = RN∗ ,

(RN)∗ = LN∗ .

Clearly all these multiplier spaces are vector subspaces of A, containing e. A partial *-algebra A is called abelian if L(a) = R(a), for all a ∈ A and a · b = b · a,

∀ b ∈ R(a).

In that case, we simply write for the multiplier spaces L(a) = R(a) ≡ M(a), LN = RN ≡ MN, (where N ⊂ A). Notice that the partial multiplication is not required to be associative (and often it is not). A partial *-algebra A is said to be associative if the following condition holds

2.1 Definitions and Examples

11

for any a, b, c ∈ A: whenever a ∈ L(b), b ∈ L(c) and a ·b ∈ L(c), then b·c ∈ R(a) and (a · b) · c = a · (b · c).

(2.1.2)

This condition is rather strong and rarely realized in practice. However, a weaker notion is sometimes useful. A partial *-algebra A is said to be semi-associative if b ∈ R(a) implies b · c ∈ R(a), for every c ∈ RA; then (2.1.2) holds. Of course, if the partial *-algebra A is semi-associative, both RA and LA are algebras.  From here on, we shall simply write ab for the product a · b.

A quasi *-algebra is a partial *-algebra for which L(a) = R(a) = A0 , if a ∈ A, and L(x) = R(x) = A, if x ∈ A0 . In other words, if (A, A0 ) is a quasi *-algebra, then A is a partial *-algebra, where the product ab of a, b ∈ A is well-defined if either a ∈ A0 or b ∈ A0 . It is clear that a quasi *-algebra is semi-associative, because of (2.1.1).

2.1.2 Basic Examples Example 2.1.3 Let A0 be a locally convex *-algebra with topology τ (the involution ∗ is supposed to be τ -continuous). If the multiplication in A is jointly continuous 0 0 τ of A0 [τ ] is a complete locally convex with respect to τ , the completion A := A *-algebra. The operations in A are defined by a limiting process. In particular, the multiplication of two elements a, b ∈ A is defined as the limit of the products of elements from two nets {xα }, {yβ } in A0 approximating respectively, a and b. If the multiplication is only separately continuous, an analogous procedure can be used only if one takes an arbitrary fixed element in A0 . More precisely, if a = limα xα , {xα } ⊂ A0 , then one can define, for each y ∈ A0 , ay = lim xα y and ya = lim yxα . α

α

0 τ , the pair (A, A0 ) is, in general, only a quasi *-algebra. This Thus, for A := A example is quite important because it was the starting point of the theory [67, 68]. Example 2.1.4 Let I = [0, 1]. Then the pair of Lebesgue spaces (Lp (I ), L∞ (I )) (with the Lebesgue measure dt on I ), 1 ≤ p < ∞, is a quasi *-algebra, with respect to the usual operations. The involution is the complex conjugation. Example 2.1.5 Let S(R) be the Schwartz space of rapidly decreasing C ∞ -functions on the real line and S  (R) the space of tempered distributions. As it is well known, S(R), which is a *-algebra with respect to the usual operations and complex conjugation, can be identified with a subspace of S  (R). The (commutative)

12

2 Algebraic Aspects

multiplication of an element of S  (R) and one of S(R) is defined as follows (φF )(ψ) = (F φ)(ψ) := F (φψ),

φ, ψ ∈ S(R), F ∈ S  (R).

It is readily checked that (S  (R), S(R)) is a quasi *-algebra. For a more detailed discussion see [73, 88]. Example 2.1.6 Let D be a dense linear subspace of a Hilbert space H and t a locally convex topology on D, finer than the topology induced by the Hilbert norm. Then, the space D× of all continuous conjugate linear functionals on D[t], i.e., the conjugate dual of D[t], is a linear vector space that contains H, in the sense that H can be identified with a subspace of D× . These identifications imply that the sesquilinear form B(·, ·) that puts D and D× in duality is an extension of the inner product of D; i.e., B(ξ, η) = ξ |η, for every ξ, η ∈ D (to simplify notation we adopt the symbol ·|· for both of them). The space D× will always be considered as endowed with the strong dual topology t × = β(D× , D). The Hilbert space H is dense in D× [t × ]. We get in this way a Gelfand triplet or rigged Hilbert space (RHS) [2, 3] D[t] → H → D× [t × ],

(2.1.3)

where → denotes a continuous embedding with dense range. As it is usual, we will systematically read (2.1.3) as a chain of inclusions and we will write D[t] ⊂ H ⊂ D× [t × ] or (D[t], H, D× [t × ]) for denoting a RHS. Let L(D, D× ) denote the vector space of all continuous linear maps from D[t] into D× [t × ]. Let D[t] → H → D× [t × ] be a rigged Hilbert space. We consider the following spaces of continuous linear maps • L(D, D× ) : the linear space of all continuous linear maps from D[t] into D× [t × ]; • L(D) : the algebra of all continuous linear maps from D[t] into itself; • L(D× ) : the algebra of all continuous linear maps from D× [t × ] into itself. Both L(D) and L(D× ) can be regarded as subspaces of L(D, D× ), in the sense that L(D, D× ) contains subspaces isomorphic to L(D) and L(D× ). If X ∈ L(D, D× ) we can define the adjoint X† of X by the equality Xξ |η = X† η|ξ ,

∀ ξ, η ∈ D.

Let us now assume that D[t] is a reflexive space so that both the topologies t and t × coincide with the corresponding Mackey topologies of the conjugate dual pair (D, D× ). In this case, we have 1. The map X → X† is an involution of L(D, D× ), i.e., X†† = X, for each X ∈ L(D, D× ). 2. L(D)† = L(D× ) and L(D) ∩ L(D× ) is a *-algebra.

2.1 Definitions and Examples

13

3. For each X ∈ L(D, D× ) and Y ∈ L(D) one has (XY )† = Y † X† . 4. For each X ∈ L(D, D× ) and Z ∈ L(D× ) one has (ZX)† = X† Z † . Let us denote by L† (D) the space of all linear operators X : D → D, having an adjoint X† : D → D, by which we simply mean that Xξ |η = ξ |X† η,

∀ ξ, η ∈ D.

The space L† (D) of continuous elements of L† (D), i.e.,   L† (D) ≡ X ∈ L† (D) : X ∈ L(D), X† ∈ L(D)

(2.1.4)

is a *-algebra and L† (D) ⊂ L(D, D× ). Moreover, L† (D) = L(D) ∩ L(D× ). Thus, from the previous considerations it follows that (L(D, D× ), L† (D)) is a quasi *-algebra.

2.1.3 Quasi *-Algebras and Partial *-Algebras of Operators Some families of unbounded operators in Hilbert spaces can be cast into the framework of quasi or partial *-algebras. This fact is important and will be used throughout this book since the partial *-algebra of operators (or partial O*-algebras [2]), we are going to introduce, is the space, where representations will take values. Let D be a dense subspace of a Hilbert space H. We denote by L† (D, H) the set of all (closable) linear operators X in H, such that D(X) = D, D(X∗ ) ⊇ D, where D(X) denotes the domain of X. The set L† (D, H) is a partial *-algebra with respect to the following operations: the usual sum X1 + X2 , the scalar multiplication λX, the involution X → X† = ∗ ∗ X∗  D and the (weak) partial multiplication X1 X2 = X1 † X2 (where X1 † ≡ (X1 † )∗ ). The latter is defined whenever X2 is a weak right multiplier of X1 (for this, we shall write X2 ∈ R w (X1 ) or X1 ∈ Lw (X2 ), that is, if and only if, X2 D ⊂ ∗ D(X1 † ) and X1† D ⊂ D(X2 ∗ ). The operator ID , restriction to D of the identity operator I on H, is the unit of the partial *-algebra L† (D, H). By L† (D, H)b we shall denote the bounded part of L† (D, H); i.e.,   L† (D, H)b = X ∈ L† (D, H) : X ∈ B(H) , where X is the closure of X, i.e., a minimal closed extension of X. Recall that when an operator X admits a closed extension is called closable and in this case, there exists a minimal closed extension of it, denoted by X and called closure of X [23, p. 28]. The space L† (D), already introduced in Example 2.1.6 can be identified with the subspace of L† (D, H) consisting of all elements in L† (D, H) that leave, together with their adjoints, the domain D invariant. It is clear that if X1 , X2 ∈

14

2 Algebraic Aspects ∗

L† (D), then (X1 X2 )ξ = X1 † X2 ξ = X1 X2 ξ , for every ξ ∈ D. Hence, L† (D) is a *-algebra with respect to the usual algebraic operations. Let L† (D)b denote the bounded part of L† (D); i.e., L† (D)b = L† (D, H)b ∩ L† (D). Then, (L† (D, H), L† (D)b ) is a quasi *-algebra. Remark 2.1.7 In general, (L† (D, H), L† (D)) is not a quasi *-algebra; indeed, if ∗ X ∈ L† (D, H) and Y ∈ L† (D), we cannot say that XD ⊂ D(Y † ). It is clear that the latter is obvious, when Y is bounded. For later use, we remind the definitions of the main topologies usually considered on L† (D, H). All these topologies are locally convex, making L† (D, H) into a locally convex space. They are defined as follows: • the weak topology tw , defined by the family of seminorms {pξ,η }, with pξ,η (X) = |Xξ |η|, X ∈ L† (D, H), ξ, η ∈ D; • the strong topology ts , defined by the family of seminorms {pξ }, with pξ (X) = Xξ , X ∈ L† (D, H), ξ ∈ D; • the strong* topology ts ∗ , defined by the family of seminorms {pξ∗ }, with pξ∗ (X) =   max Xξ , X† ξ  , X ∈ L† (D, H), ξ ∈ D. Clearly, ts ∗ is finer than ts , which in turn is finer than tw , in general. The involution * is continuous with respect to tw and ts ∗ , while none of these topologies makes the weak multiplication continuous.

2.2 Homomorphisms, Ideals and Representations Definition 2.2.1 Let (A, A0 ) be a quasi *-algebra and M a subspace of A. Consider the conditions: (i) a ∈ M ⇔ a ∗ ∈ M; (ii) xa ∈ M, for every a ∈ M, x ∈ A0 ; if (i) and (ii) hold, then M is called an A0 -*-submodule of A. If, in addition, A0 ⊂ M, then (M, A0 ) is called a quasi *-subalgebra of (A, A0 ). Remark 2.2.2 Clearly (i) and (ii) of the previous definition imply that ax ∈ M, for every a ∈ M, x ∈ A0 . Moreover, if (A, A0 ) has a quasi-unit q, then for every A0 -*-submodule M of A, such that q ∈ M, (M, A0 ) is a quasi *-subalgebra of (A, A0 ). Definition 2.2.3 Let (A, A0 ) be a quasi *-algebra and B a partial *-algebra. A linear map from A into B is called a homomorphism of (A, A0 ) into B if, for a ∈ A and x ∈ A0 , (a) (x) and (x) (a) are well defined in B and

(a) (x) = (ax), (x) (a) = (xa), respectively. The homomorphism is called a *-homomorphism if (a ∗ ) = (a)∗ , for every a ∈ A.

2.2 Homomorphisms, Ideals and Representations

15

If (B, B0 ) is a quasi *-algebra and is a homomorphism (resp., *-homomorphism) of (A, A0 ) into B, then the above definition implies that ( (a), (b)) ∈ B × B0 ∪ B0 × B, whenever a, b ∈ A and either a ∈ A0 or b ∈ A0 . This in turn implies that

(A0 ) ⊆ B0 . For this reason, we call a qu-homomorphism (resp., qu*-homomorphism). If is a qu*-homomorphism, ( (A), (A0 )) is a quasi *-subalgebra of (B, B0 ). Definition 2.2.4 Let (A, A0 ) be a quasi *-algebra and I a subspace of A. We say that I is a left qu-ideal if ax ∈ I, for every x ∈ I and a ∈ L(x). Similarly, I is a right qu-ideal if xa ∈ I, for every x ∈ I and a ∈ R(x). Finally, I is a (two-sided) qu-ideal if it is both a left and right qu-ideal. A left or right qu-ideal I is a qu*-ideal if a ∈ I implies a ∗ ∈ I. A qu*-ideal is necessarily a two-sided qu-ideal. The kernel of a *-homomorphism is clearly a qu*-ideal. If I is a left qu-ideal, then I is a left A0 -module, since A0 ⊂ LI. Let I be a qu*-ideal of A, such that I ⊂ RI. Then, the quotient A/I, which is a vector space with the usual sum, can be made into a partial *-algebra by setting    = (a + I, b + I) : a ∈ L(b), a ∈ LI, b ∈ RI , (a + I) · (b + I) = ab + I, (a + I)∗ = a ∗ + I. Definition 2.2.5 Let (A, A0 ) be a quasi *-algebra and Dπ a dense domain in a certain Hilbert space Hπ . A linear map π from A into L† (Dπ , Hπ ) is called a *-representation of (A, A0 ), if the following properties are fulfilled: (i) π(a ∗ ) = π(a)† , ∀ a ∈ A; (ii) for a ∈ A and x ∈ A0 , π(a)π(x) is well defined and π(a)π(x) = π(ax). In other words, π is a *-homomorphism of (A, A0 ) into the partial *-algebra L† (Dπ , Hπ ). If (A, A0 ) has a unit e ∈ A0 , we assume π(e) = IDπ , where IDπ is the identity operator on the space Dπ . If πo := π A0 is a *-representation of the *-algebra A0 into L† (Dπ ) we say that π is a qu*-representation. If π is a *-representation of (A, A0 ), then the closure  π of π is defined, for each π , which is the completion of a ∈ A, as the restriction of π(a) to the domain D Dπ under the graph topology tπ [12, p. 9] defined by the seminorms ξ ∈ Dπ → ξ  + π(a)ξ , a ∈ A, where  ·  is the norm induced by the inner product on Dπ . If π =  π , the *-representation is said to be closed. The adjoint of a *-representation π of a quasi *-algebra (A, A0 ), denoted by π ∗ , is defined as follows; see [2, 23] Dπ ∗ ≡

 a∈A

D(π(a)∗ ) and π ∗ (a) = π(a ∗ )∗ Dπ ∗ ,

a ∈ A.

16

2 Algebraic Aspects

The *-representation π is said to be selfadjoint if π = π ∗ . Finally, a *-representation π is called bounded if π(a) is a bounded operator in Dπ , for every a ∈ A. The *-representation π is said to be • ultra-cyclic, if there exists ξ0 ∈ Dπ , such that Dπ = π(A0 )ξ0 ; • strongly-cyclic, if there exists ξ0 ∈ Dπ , such that π(A0 )ξ0 is dense in Dπ with respect to the graph topology tπ ; • cyclic, if there exists ξ0 ∈ Dπ , such that π(A0 )ξ0 is dense in H in its norm topology.   Let πα : α ∈ I be a family of closed *-representations of A. The algebraic direct sum of the *-representations πα ’s, denoted by π , is defined on the pre-Hilbert space Dπ =



 Dπα := (ξα )α∈I ∈ α∈I Dπα , with ξα = 0, for at most finitely

α∈I

many α ∈ I .

Inner product on Dπ and π(x) on Dπ , x ∈ A, are given as follows   ξα , ηα α , ∀ (ξα )α∈I , (ηα )α∈I ∈ D and (ξα )α∈I ), (ηα )α∈I ) :=





π(x) (ξα )α∈I

α∈I



:= πα (x)ξα α∈I , ∀ x ∈ A and (ξα )i∈I ∈ Dπ .

Considering now the Hilbert spaces Hα ’s, completions of the pre-Hilbert spaces Dα ’s, α ∈ I , we take the Hilbert space direct sum of Hα ’s, given by   πα (a)2 < ∞, ∀ a ∈ A . Hπ = ξ = (ξα )α∈I : ξα ∈ Dπα , α∈I

We clearly have that Dπ is dense in Hπ . Now, the closure  π , of the *-representation algebraic direct sum π , is called direct sum of the family of the closed *representations πα : α ∈ I , and it is denoted by ⊕α∈I πα . Following [23], we say that a subspace M of Dπ is invariant for a *representation π of (A, A0 ) if π(a)ξ ∈ M, for every a ∈ A and ξ ∈ M. A closed subspace K of Hπ is called invariant for π if there exists a subspace M of Dπ , which is dense in K and invariant for π . If M ⊂ Dπ is invariant for π , then the mapping a → π(a) M defines a *-representation πM of (A, A0 ) and clearly M is invariant for πM . We say that a subspace M of Dπ is reducing for π if there exist *-representations π1 , π2 of A, with π1 having M as domain, such that π = π1 ⊕ π2 . As for bounded representations, reducing subspaces can be described in terms of projections in the commutant of the image of the *-representation involved.

2.3 Families of Sesquilinear Forms

17

Let D be a subspace of a Hilbert space H. If Y is a † -invariant subset of the weak bounded commutant of Y is defined to be the set

L† (D, H),

 (Y, D)w = B ∈ B(H) : BY ξ |η = Bξ |Y ∗ η,

 ∀ Y ∈ Y, ξ, η ∈ D .

It is easily seen that (Y, D)w is a weakly closed ∗ -invariant subspace of B(H). One also defines the strong bounded commutant of Y as   (Y, D)s = B ∈ (Y, D)w : BD ⊂ D . It is easily seen that (Y, D)s is an algebra, but it is not ∗ -invariant, in general. Then, one has the following (for the relevant terminology, see [23, p. 214, Lemma 8.3.5]). Proposition 2.2.6 Let π be a *-representation of A defined on Dπ and let M be a subspace of Dπ . Denote by PM the projection onto the closure M of M in H. The following statements are equivalent: (i) M is reducing for π . (ii) M and M⊥ ∩ Dπ are both invariant for π and PM Dπ ⊂ Dπ . (iii) PM ∈ (π(A), Dπ )s . A *-representation π of A, defined on Dπ , is said to be irreducible if there is no nontrivial subspace M of Dπ , which is reducing for π . By Proposition 2.2.6 it follows immediately that π is irreducible, if and only if, (π(A), Dπ )s = C I.

2.3 Families of Sesquilinear Forms Definition 2.3.1 Let (A, A0 ) be a quasi *-algebra. We denote with QA0 (A) the set of all sesquilinear forms on A × A, such that (i) ϕ is positive, i.e., ϕ(a, a) ≥ 0, ∀ a ∈ A; (ii) ϕ(ax, y) = ϕ(x, a ∗ y), ∀ a ∈ A, x, y ∈ A0 . Let ϕ ∈ QA0 (A). Then, the positivity of ϕ implies that ϕ(a, b) = ϕ(b, a),

∀ a, b ∈ A;

|ϕ(a, b)|2 ≤ ϕ(a, a)ϕ(b, b),

∀ a, b ∈ A.

Hence,     Nϕ := a ∈ A : ϕ(a, a) = 0 = a ∈ A : ϕ(a, b) = 0, ∀ b ∈ A ,

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2 Algebraic Aspects

so Nϕ is a subspace with the property that, if x ∈ Nϕ ∩ A0 then ϕ(ax, y) = 0,

∀ y ∈ A0 ,

but it is not necessarily a left qu-ideal of A. Let λϕ : A → A/Nϕ be the usual quotient map and for each a ∈ A, let λϕ (a) be the corresponding coset of A/Nϕ , which contains a. We define an inner product ·|· on λϕ (A) = A/Nϕ by λϕ (a)|λϕ (b) := ϕ(a, b),

∀ a, b ∈ A.

(2.3.5)

Denote by Hϕ the Hilbert space obtained by the completion of the pre-Hilbert space λϕ (A). Proposition 2.3.2 Let ϕ ∈ QA0 (A). The following statements are equivalent: (i) λϕ (A0 ) is dense in Hϕ . (ii) If {an } is a sequence of elements of A such that: (ii.a) ϕ(an , x) → 0, as n → ∞, for every x ∈ A0 ; (ii.b) ϕ(an − am , an − am ) → 0, as n, m → ∞; then, lim ϕ(an , an ) = 0. n→∞

Proof (i) ⇒ (ii) Let {an } be a sequence of elements of A for which (ii.a) and (ii.b) hold. By (ii.b) the sequence {λϕ (an )} is Cauchy in Hϕ . Let ξ be its limit. By (ii.a) it follows that ξ |λϕ (x) = lim ϕ(an , x) = 0, n→∞

∀ x ∈ A0 .

Hence, ξ is orthogonal to λϕ (A0 ). This implies that ξ = 0 and, therefore, lim ϕ(an , an ) = ξ 2 = 0.

n→∞

(ii) ⇒ (i) Let ξ ∈ Hϕ be a vector orthogonal to λϕ (A0 ) and {an } a sequence in A such that λϕ (an ) → ξ . Then, it is easily seen that {an } satisfies (ii.a) and (ii.b). Moreover, ξ 2 = lim ϕ(an , an ) = 0. n→∞

This proves that λϕ (A0 ) is dense in Hϕ .

 

We denote by IA0 (A) the subset of all sesquilinear forms ϕ ∈ QA0 (A) satisfying (i) or (ii) of Proposition 2.3.2. Remark 2.3.3 Elements of IA0 (A) are instances of invariant positive sesquilinear forms (ips-forms, for short) as defined in [39, p. 5] for partial *-algebras. The elements ϕ ∈ IA0 (A) are characterized by the existence of a subspace B(ϕ) of

2.4 Construction of *-Representations

19

A (called a core for ϕ), whose properties are essentially the same as those that A0 possesses, in view of Definition 2.3.1 and Proposition 2.3.2. In fact, in the definition of an ips-form, there is an additional condition, which takes into account the possible lack of associativity in a partial *-algebra. In particular, an ips-form is an everywhere defined biweight in the sense of [2]. Elements of IA0 (A) are nothing but ips-forms on A with fixed core the *-algebra A0 . The opposite of ips-forms are singular forms, which we define as follows [53, 82]: Definition 2.3.4 Let (A, A0 ) be a quasi *-algebra and ψ ∈ QA0 (A). We say that ψ is A0 -singular if there exists a0 ∈ A with ψ(a0 , a0 ) > 0 and ψ(a, x) = 0, for every a ∈ A, x ∈ A0 . Example 2.3.5 In general, QA0 (A)  IA0 (A) and therefore singular forms do really exist. We construct such an example as follows. Let B be a C*-algebra with norm  ·  and unit e. Let J be a proper dense *-ideal of B. Let v0 ∈ B, v0 = v0∗ , be an element of B satisfying the following condition αe + βv0 + γ v02 ∈ J ⇔ α = β = γ = 0.

(2.3.6)

Let   A := λe + μv0 + u : λ, μ ∈ C, u ∈ J . From (2.3.6) it follows that for every a ∈ A the decomposition a = λa e + μa v0 + ua with λa , μa ∈ C and ua ∈ J is unique. Put   A0 = λe + u : λ ∈ C, u ∈ J . From (2.3.6) it also follows that there is no element x ∈ A0 , such that v0 x = v02 , hence (A, A0 ) is a true quasi *-algebra with respect to the multiplication inherited on A0 by the C*-algebra B. Now we define ϕ(λa e + μa v0 + ua , λb e + μb v0 + ub ) = μa μb ω(v0∗ v0 ) where ω is a positive linear functional on B, such that ω(v0∗ v0 ) > 0. Then, it is easily seen that ϕ ∈ QA0 (A). But ϕ ∈ IA0 (A), since Nϕ = A0 and so A/Nϕ = C (up to an algebraic isomorphism), while λϕ (A0 ) = {0}.

2.4 Construction of *-Representations The Gelfand–Naimark–Segal (GNS) construction for positive linear functionals is one of the most relevant tools for the study of (locally convex) *-algebras [9, 19, 23]. As customary, when a partial multiplication is involved [2], we consider, as starting

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2 Algebraic Aspects

point for the construction, a positive sesquilinear form enjoying certain invariance properties (see, also, [49]). For a quasi *-algebra (A, A0 ) this set of sesquilinear forms is exactly IA0 (A). An analogous GNS construction for positive linear functionals is also possible, under certain circumstances, and it will be discussed later (Theorem 2.4.8).

2.4.1 GNS-Like Construction with Sesquilinear Forms Proposition 2.4.1 Let (A, A0 ) be a quasi *-algebra with unit e and ϕ a sesquilinear form on A × A. The following statements are equivalent: (i) ϕ ∈ IA0 (A). (ii) There exist a Hilbert space Hϕ , a dense domain Dϕ of the Hilbert space Hϕ and a closed cyclic *-representation πϕ in L† (Dϕ , Hϕ ), with cyclic vector ξϕ (in the sense that πϕ (A0 )ξϕ is dense in Hϕ ), such that ϕ(a, b) = πϕ (a)ξϕ |πϕ (b)ξϕ ,

∀ a, b ∈ A.

(2.4.7)

Proof (i) ⇒ (ii) Let ϕ ∈ IA0 (A). Put πϕ◦ (a)λϕ (x) := λϕ (ax),

a ∈ A, x ∈ A0 .

(2.4.8)

First we prove that, for every a ∈ A, the map πϕ◦ (a) is well-defined. Assume that, for x ∈ A0 , λϕ (x) = 0. If a ∈ A, we then get ϕ(x, a ∗ y) = 0, for every y ∈ A0 . Since ϕ ∈ IA0 (A), for each b ∈ A, there exists a sequence yn ∈ A0 , such that λϕ (b) − λϕ (yn ) → 0, as n → ∞. This clearly implies that ϕ(ax, b) = 0, for every b ∈ A. Hence, ax ∈ Nϕ . Thus, for every a ∈ A, the map πϕ◦ (a) is a welldefined linear operator from λϕ (A0 ) into Hϕ . This fact together with the properties of ϕ listed in Definition 2.3.1 easily imply that πϕ◦ is a *-representation, whose restriction to A0 , maps λϕ (A0 ) into itself. Since (A, A0 ) has a unit e, then (i) and (ii) follow from the very definitions. Denote by πϕ the closure of πϕ◦ and by Dπϕ ≡ Dϕ its domain. Then, it is easily seen that πϕ satisfies (2.4.7). It is also clear by the definition of πϕ that, since (A, A0 ) has a unit e, then ξϕ := λϕ (e) is a cyclic vector for πϕ . (ii) ⇒ (i) From the equality (2.4.7) follows easily that ϕ(a, a) ≥ 0,

∀ a ∈ A;

ϕ(ax, y) = ϕ(x, a ∗ y),

∀ a ∈ A, x, y ∈ A0 .

2.4 Construction of *-Representations

21

Since πϕ (A0 )ξϕ is dense in Hϕ , for every a ∈ A, there exists a sequence {xn }, n ∈ N, in A0 , such that πϕ (a)ξϕ − πϕ (xn )ξϕ  → 0, as n → ∞. Therefore, λϕ (a) − λϕ (xn )2 = ϕ(a − xn , a − xn ) = πϕ (a)ξϕ − πϕ (xn )ξϕ 2 → 0, as n → ∞. This implies that λϕ (A0 ) is dense in Hϕ ; i.e., ϕ ∈ IA0 (A).

 

Definition 2.4.2 The triple (πϕ , λϕ , Hϕ ) constructed in Proposition 2.4.1 is called the GNS construction for ϕ and πϕ is called the GNS representation of A corresponding to ϕ. Proposition 2.4.3 Let (A, A0 ) be a quasi *-algebra with unit e and ϕ ∈ IA0 (A). Then, the GNS construction (πϕ , λϕ , Hϕ ) is unique up to unitary equivalence. Proof Let (πϕ , λϕ , Hϕ ) be another triple satisfying the same conditions as (πϕ , λϕ , Hϕ ). Like above, we put ξϕ = λϕ (e) and define U πϕ (a)ξϕ := πϕ (a)ξϕ ,

a ∈ A.

Then, U πϕ (a)ξϕ |U πϕ (b)ξϕ  = πϕ (a)ξϕ |πϕ (b)ξϕ  = ϕ(a, b) = πϕ (a)ξϕ |πϕ (b)ξϕ , for all a, b ∈ A, which proves, at once, that U : Dϕ → Dϕ is well-defined and preserves the inner product. Clearly U extends to a unitary operator from Hϕ onto Hϕ , which we denote by the same symbol. With Dπϕ ≡ Dϕ and Dπ ϕ ≡ Dϕ , it is easily seen that U Dπϕ = Dπ ϕ and U πϕ (a)η = πϕ (a)U η,

∀ a ∈ A, η ∈ Dπϕ .

Hence, the two *-representations πϕ and πϕ are unitarily equivalent.

 

Definition 2.4.4 A sesquilinear form ϕ is called admissible if, for every a ∈ A, there exists γa > 0, such that ϕ(ax, ax) ≤ γa ϕ(x, x),

∀ x ∈ A0 .

Proposition 2.4.5 The *-representation πϕ is bounded, if and only if, ϕ is admissible. Proof From the construction in Proposition 2.4.1, we obtain the equality πϕ (a)λϕ (x) = λϕ (ax) = ϕ(ax, ax)1/2 ,

∀ a ∈ A, x ∈ A0 .

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2 Algebraic Aspects

Thus, if πϕ is bounded, then for every a ∈ A, there exists γa > 0, such that ϕ(ax, ax) = π(a)λϕ (x)2 ≤ γa λϕ (x) = γa ϕ(x, x),

∀ x ∈ A0 .  

The converse is proved in a similar way.

2.4.2 GNS-Like Construction with Linear Functionals As announced at the beginning of this section, certain linear functionals over a quasi *-algebra (A, A0 ) allow a GNS-like construction. Definition 2.4.6 Let (A, A0 ) be a quasi *-algebra and ω be a linear functional on A satisfying the following conditions: (L.1) ω(x ∗ x) ≥ 0, ∀ x ∈ A0 ; (L.2) ω(y ∗ a ∗ x) = ω(x ∗ ay), ∀ a ∈ A, x, y ∈ A0 ; (L.3) ∀ a ∈ A, there exists γa > 0, such that |ω(a ∗ x)| ≤ γa ω(x ∗ x)1/2 ,

∀ x ∈ A0 .

Then, ω is called representable linear functional on A. The family of representable linear functionals on (A, A0 ) is denoted by R(A, A0 ). Remark 2.4.7 The family R(A, A0 ) enjoys the following properties: (a) If ω1 , ω2 ∈ R(A, A0 ), then ω1 + ω2 ∈ R(A, A0 ), as well as λω1 ∈ R(A, A0 ), for every λ ≥ 0. Indeed, the conditions (L.1) and (L.2) are obviously satisfied. As for (L.3), for every a ∈ A, there exist γa,1 , γa,2 > 0, such that |ω1 (a ∗ x)| ≤ γa,1 ω1 (x ∗ x)1/2 , |ω2 (a ∗ x)| ≤ γa,2 ω2 (x ∗ x)1/2 ,

∀ x ∈ A0 ,

hence, |(ω1 + ω2 )(a ∗ x)| ≤ γa,1 ω1 (x ∗ x)1/2 + γa,2 ω2 (x ∗ x)1/2 

 ≤ max γa,1 , γa,2 ω1 (x ∗ x)1/2 + ω2 (x ∗ x)1/2 √ 

1/2  . ≤ 2 max γa,1 , γa,2 (ω1 + ω2 )(x ∗ x) (b) If ω ∈ R(A, A0 ) and z ∈ A0 , then the linear functional ωz , defined by ωz (a) := ω(z∗ az), is representable. We omit the easy proof.

a ∈ A,

2.4 Construction of *-Representations

23

Theorem 2.4.8 Let (A, A0 ) be a quasi *-algebra with unit e and let ω be a representable linear functional on A. Then, there exists a closed cyclic *-representation πω of (A, A0 ), with a cyclic vector ηω , such that ω(a) = πω (a)ηω |ηω ,

∀ a ∈ A.

This representation is unique up to unitary equivalence.   Proof We define Nω = x ∈ A0 : ω(x ∗ x) = 0 . Then, Nω is a left-ideal of A0 and the quotient A0 /Nω is a pre-Hilbert space with inner product λω (x)|λω (y) = ω(y ∗ x),

x, y ∈ A0 ,

where λω (x), x ∈ A0 , denotes the coset in A0 /Nω containing x. Let Hω be the completion of λω (A0 ). If a ∈ A, we put a ω (λω (x)) = ω(a ∗ x). Then, by (L.3), it follows that a ω is a well defined linear functional on λω (A0 ) and we have

|a ω λω (x) | = |ω(a ∗ x)| ≤ γa ω(x ∗ x)1/2 = γa λω (x),

∀ x ∈ A0 .

Thus, a ω is bounded and by Riesz’s lemma, there exists a unique ξωa ∈ Hω , such that

ω(a ∗ x) = a ω λω (x) = λω (x)|ξωa ,

∀ x ∈ A0 .

(2.4.9)

Furthermore, we put πω◦ (a)λω (x) := ξωax ,

x ∈ A0 .

(2.4.10)

Since, λω (y)|πω◦ (a)λω (x) = λω (y)|ξωax  = (ax)ω (λω (y)) = ω(x ∗ a ∗ y) = ω(y ∗ ax),

∀ y ∈ A0 ,

it follows from (L.3) that πω◦ (a) is well-defined and maps λω (A0 ) into Hω . In a similar way one can show the equality πω◦ (a ∗ )λω (y)|λω (x) = ω(y ∗ ax),

∀ x, y ∈ A0 .

This implies that πω◦ (a) ∈ L† (λω (A0 ), Hω ) and πω◦ (a)† = πω◦ (a ∗ ), for all a ∈ A. By the uniqueness of ξωa ∈ Hω , we may extend the map λω on A, keeping the same symbol; i.e., λω : A → Hω : a → λω (a) := ξωa .

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2 Algebraic Aspects

Then, from (2.4.10) we obtain πω◦ (x)λω (y) = λω (xy),

∀ x, y ∈ A0 ,

(2.4.11)

With analogous computations as before and taking into account the preceding equality, we conclude that πω◦ (ax)λω (y)|λω (z) = πω◦ (x)λω (y)|πω◦ (a ∗ )λω (z),

∀ y, z ∈ A0 .

This implies that πω◦ (a)πω◦ (x) (see beginning of Sect. 2.1.3) is well-defined and πω◦ (ax) = πω◦ (a)πω◦ (x),

∀ a ∈ A, x ∈ A0 .

Thus, πω◦ is a *-representation. It is clear that πω◦ (A0 )ηω , ηω := λω (e), is dense in Hϕ . Taking, finally, the closure πω of πω◦ we get the desired closed *-representation. The statement about the uniqueness comes again from a slight modification of the classical argument.    ◦ ∗ a D(πω (x) ) and Remark 2.4.9 Note that, for every a ∈ A, ξω ∈ D(πω◦ )∗ = x∈A0

(πω◦ )∗ (x)ξωa = ξωxa , for every x ∈ A0 . Indeed, from (2.4.11) and (2.4.9), we obtain λω (y)|(πω◦ )∗ (x)ξωa  = πω◦ (x ∗ )λω (y)|ξωa 



= λω (x ∗ y)|ξωa  = ω (xa)∗ y = λω (y)|ξωxa ,

∀ y ∈ A0 .

So the proof of our claim is completed. The *-representation πω satisfies, therefore, the following properties: πω (a)λω (x) = ξωax and πω∗ (x)ξωa = ξωxa ,

∀ a ∈ A, x ∈ A0 .

Remark 2.4.10 The *-representation πω is bounded, if and only if, for every a ∈ A there exists γa > 0, such that |ω(y ∗ ax)| ≤ γa ω(x ∗ x)1/2 ω(y ∗ y)1/2 ,

∀ x, y ∈ A0 .

Remark  2.4.11 It is not difficult to show that, for every a ∈ A, πω (a) maps λω (A0 ) into x∈A0 D(πω (x)∗ ). The GNS construction given above also implies the following Proposition 2.4.12 Let (A, A0 ) be a quasi *-algebra with unit e and ω ∈ R(A, A0 ). Then, there exists a linear operator Tω : A → Hω , such that ω(a) = Tω a|ξω ,

∀ a ∈ A,

where ξω is the cyclic vector of the GNS representation associated to ω.

2.4 Construction of *-Representations

25

Proof It is enough to put Tω a := ξωa , as defined in the proof of Theorem 2.4.8.

 

Example 2.4.13 Let D be a dense domain in a Hilbert space H and  · 1 a norm on D, stronger than the Hilbert norm  · . Let B(D, D) denote the vector space of all jointly continuous sesquilinear forms on D × D, with respect to  · 1 . The map ϕ → ϕ ∗ , with ϕ ∗ (ξ, η) := ϕ(η, ξ ), defines an involution in B(D, D). As in (2.1.4), L† (D) denotes the *-subalgebra of L† (D) consisting of all operators A ∈ L† (D) such that both A and A† are continuous from D[ · 1 ] into itself. Every A ∈ L† (D) defines a sesquilinear form ϕA ∈ B(D, D) by ϕA (ξ, η) := Aξ |η,

ξ, η ∈ D.

Indeed, for every ξ, η ∈ D, there exists γ > 0, such that |ϕA (ξ, η)| = |Aξ |η| ≤ Aξ η ≤ γ Aξ 1 η1 ≤ γ  ξ 1 η1 . We put   B† (D) := ϕA : A ∈ L† (D) . ∗ = ϕ , for every A ∈ L† (D). Moreover, if ϕ ∈ B(D, D), It is easily seen that ϕA A† † then ϕA ∈ B (D). We now define

(ϕ ◦ ϕA )(ξ, η) := ϕ(Aξ, η), (ϕA ◦ ϕ)(ξ, η) := ϕ(ξ, A† η),

ξ, η ∈ D, ξ, η ∈ D.

Under the involution and the operations defined before, the pair (B(D, D), B† (D)) is a quasi *-algebra (see, also [2, Chap. 10], for a complete discussion). For every ξ ∈ D, we define ωξ (ϕ) := ϕ(ξ, ξ ),

ϕ ∈ B(D, D).

Then, ωξ is a linear functional on B(D, D). Moreover, ωξ (ϕA† ◦ ϕA ) = (ϕA ◦ ϕA )(ξ, ξ ) = Aξ |Aξ  ≥ 0. ωξ (ϕB † ◦ ϕ ◦ ϕA ) = ϕ(Aξ, Bξ ) = ωξ (ϕA† ◦ ϕ ∗ ◦ ϕB† ). Hence, ωξ satisfies (L.1) and (L.2).

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2 Algebraic Aspects

The functional ωξ satisfies (L.3), if and only if, for every ϕ ∈ B(D, D), there exists γϕ > 0, such that |ϕ(Aξ, ξ )| ≤ γϕ Aξ ,

∀ A ∈ L† (D).

Indeed, ∗ ◦ ϕA )1/2 . |ωξ (ϕ ∗ ◦ ϕA )| = |(ϕ ∗ ◦ ϕA )(ξ, ξ )| = |ϕ(Aξ, ξ )| ≤ γϕ Aξ  = γϕ ωξ (ϕA

The previous condition is clearlysatisfied, if and only  if, ϕ is bounded in the first variable on the subspace Mξ = Aξ : A ∈ L† (D) . If this is the case, then there exists ζ ∈ H, such that ωξ (ϕ ◦ ϕA ) = Aξ |ζ ,

∀ A ∈ L† (D).

Hence, not every ωξ is representable. Let now ϕ ∈ IA0 (A); then the linear functional ωϕ , with ωϕ (a) := ϕ(a, e), a ∈ A, satisfies the conditions (L.1), (L.2) and (L.3) of Definition 2.4.6, therefore it is representable. Thus, Theorem 2.4.8 can be applied to get the *-representation  πωϕ constructed as shown above. On the other hand, we can also build up, as in Proposition 2.4.1, the closed *-representation πϕ , with cyclic vector ξϕ . Since ωϕ (a) = ϕ(a, e) = πϕ (a)ξϕ |ξϕ ,

∀ a ∈ A,

it turns out that  πωϕ and πϕ are unitarily equivalent. Let ϕ ∈ QA0 (A). Then, the linear functional ωϕ , with ωϕ (a) = ϕ(a, e), a ∈ A, is representable. Let πωϕ be the corresponding *-representation. If we define ϕ (a, b) := πωϕ (a)ξωϕ |πωϕ (b)ξωϕ ,

a, b ∈ A,

it is easily seen that ϕ ∈ QA0 (A). But, in general, ϕ = ϕ. Proposition 2.4.14 The following statements hold: (i) for every ϕ ∈ QA0 (A), ϕ ∈ IA0 (A); (ii) for every ϕ ∈ QA0 (A), there exists ϕ0 ∈ IA0 (A), which coincides with ϕ on all pairs (a, x) with a ∈ A, x ∈ A0 and an A0 -singular form sϕ (see Definition 2.3.4), such that ϕ(a, b) = ϕ0 (a, b) + sϕ (a, b),

∀ a, b ∈ A,

where ϕ0 = ϕ and sϕ = ϕ − ϕ ; (iii) ϕ = ϕ, if and only if, λϕ (A0 ) is dense in Hϕ , i.e., if and only if, ϕ ∈ IA0 (A).

2.4 Construction of *-Representations

27

Proof (i) Since πωϕ is cyclic, then for every a ∈ A, there exists a sequence {xn } ⊂ A0 , such that πωϕ (a − xn )ξωϕ  → 0, as n → ∞. Then, we have λϕ (a) − λϕ (xn )2 = πωϕ (a − xn )ξωϕ |πωϕ (a − xn )ξωϕ  → 0. n→∞

This implies that λϕ (A0 ) is dense in Hϕ . (ii) Put ϕ0 = ϕ and sϕ = ϕ −ϕ . Clearly, sϕ (a, x) = 0, for every a ∈ A, x ∈ A0 . Now we prove that ϕ (a, a) ≤ ϕ(a, a), for every a ∈ A. Indeed, we have ϕ (a, a) = πωϕ (a)λωϕ (e)2 = λωϕ (a)2 . Notice that, by the construction in Theorem 2.4.8,   λωϕ (a) = sup |ωϕ (a ∗ x)| : ωϕ (x ∗ x) = 1   = sup |ϕ(a, x)| : ωϕ (x ∗ x) = 1 ≤ ϕ(a, a)1/2 . Hence, sϕ ∈ QA0 (A). If λϕ (A0 ) is not dense in Hϕ , there exists a sequence {an } of elements of A with the properties (see Proposition 2.3.2): (a) ϕ(an , x) → 0, n→∞

∀ x ∈ A0 ;

(b) ϕ(an − am , an − am ) → 0; n→∞

(c) lim ϕ(an , an ) = α > 0. n→∞

Since λϕ (A) is dense in Hϕ and ϕ (an − am , an − am ) ≤ ϕ(an − am , an − am ) → 0, we have limn→∞ ϕ (an , an ) = 0. In conclusion, sϕ (an , an ) → α > 0. So n→∞

that sϕ cannot be identically 0. This clearly implies that sϕ is A0 -singular. (iii) is clear.

 

If π is a *-representation of A then, for every ξ ∈ Dπ , the vector form ϕξ defined by ϕξ (a, b) := π(a)ξ |π(b)ξ ,

a, b ∈ A,

(2.4.12)

is an element of QA0 (A), but it need not belong to IA0 (A). For this reason, we say that π is regular if, for every ξ ∈ Dπ , ϕξ ∈ IA0 (A).

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2 Algebraic Aspects

Proposition 2.4.15 Let π be a *-representation of (A, A0 ). The following statements are equivalent: (i) π is regular; (ii) π(a)ξ ∈ π(A0 )ξ , for every a ∈ A and for every ξ ∈ Dπ ; (iii) for every ξ ∈ Dπ , π0 := π Mξ is a *-representation of (A, A0 ) into L† (Mξ , Mξ ), where Mξ = π(A0 )ξ (i.e., Mξ is an invariant subspace for π ). Proof (i) ⇒ (ii) Let π be regular and ξ ∈ Dπ . Then, ϕξ ∈ IA0 (A), hence (Proposition 2.3.2) for every a ∈ A, there exists a sequence {xn } ⊂ A0 , such that λϕξ (a − xn ) → 0. Consequently, (π(a) − π(xn ))ξ 2 = λϕξ (a − xn )2 → 0. This proves that π(a)ξ ∈ π(A0 )ξ . (ii) ⇒ (iii) The assumption (ii) implies that, for every a ∈ A and ξ ∈ Dπ , π0 (a) maps π(A0 )ξ into π(A0 )ξ . Some simple calculations, which make use of the fact that π is a *-representation and that the module associativity holds, show that π0 (a ∗ ) = (π0 (a))∗ Mξ and that π0 preserves the partial multiplication of (A, A0 ). (iii) ⇒ (i) The assumption (iii) yields that, for every ξ ∈ Dπ and a ∈ A, π(a)ξ ∈ Mξ . Therefore, for every a ∈ A, there exists a sequence {xn } ⊂ A0 such that (π(a) − π(xn ))ξ  → 0. Then, for ϕξ , we have ϕξ (a − xn , a − xn ) = λϕξ (a − xn )2 = (π(a) − π(xn ))ξ 2 → 0.  

Consequently, π is regular.

If ϕ is a sesquilinear form on A × A and x ∈ A0 , we denote with ϕx the sesquilinear form on A × A defined by ϕx (a, b) := ϕ(ax, bx),

a, b ∈ A,

(2.4.13)

and with ωϕx the corresponding linear functional on A defined by ωϕx (a) := ϕ(ax, x),

a ∈ A.

(2.4.14)

It is readily checked that, if ϕ ∈ QA0 (A), then ϕx ∈ QA0 (A) too, for all x ∈ A0 . Proposition 2.4.16 Let (A, A0 ) be a quasi *-algebra with unit e, ϕ ∈ IA0 (A) and πϕ◦ the *-representation defined in (2.4.8). The following statements are equivalent: (i) πϕ◦ is regular; (ii) ϕx ∈ IA0 (A),

∀ x ∈ A0 .

2.4 Construction of *-Representations

29

Proof If η ∈ Dϕ = λϕ (A0 ), then η = λϕ (x), for some x ∈ A0 . Hence, ϕη (a, b) = πϕ◦ (a)λϕ (x)|πϕ◦ (b)λϕ (x) = ϕ(ax, bx) = ϕx (a, b),

∀ a, b ∈ A.

Thus, ϕη = ϕx . This equality clearly implies the equivalence of (i) and (ii).

 

It is then useful to introduce the notation  s (A) := ϕ ∈ IA0 (A) : ϕx ∈ IA0 (A), IA 0

 ∀ x ∈ A0 .

s (A), the corresponding *-representation π ◦ is It is clear that for every ϕ ∈ IA ϕ 0 regular.

Remark 2.4.17 For the GNS representation πϕ constructed by ϕ (i.e., for the closure of πϕ◦ ) the implication (i) ⇒ (ii) still holds, in an obvious way; however, (ii) does not imply (i), in general. Example 2.4.18 Let π be a regular *-representation of (A, A0 ) in L† (Dπ , Hπ ). Let ξ ∈ Dπ and let ϕξ be the corresponding vector form (in the sense of (2.4.12)). Then, by definition, ϕξ ∈ IA0 (A). Since, π(x)ξ ∈ Dπ , for every x ∈ A0 , one also s (A). has (ϕξ )x ∈ IA0 (A), for every x ∈ A0 . In this case, ϕξ ∈ IA 0 Notice that a bounded *-representation π of A (i.e., π(a) ∈ B(H), for every a ∈ A) need not be regular.  In the sequel, we shall indicate by Rep(A), Repr (A)

the family of all *representations and all regular *-representations of (A, A0 ), respectively.

Chapter 3

Normed Quasi *-Algebras: Basic Theory and Examples

In this chapter we shall consider the case, where A is endowed with a norm topology, making (A, A0 ) into a normed quasi *-algebra in the sense of Definition 3.1.1, below. This opens our discussion on locally convex quasi *-algebras, starting from the simplest situation. Nevertheless, as we shall see, simple does not mean trivial at all.

3.1 Basic Definitions and Facts Definition 3.1.1 A quasi *-algebra (A, A0 ) is called a normed quasi *-algebra if A is a normed space under a norm  ·  satisfying the following properties: (i) a ∗  = a, ∀ a ∈ A; (ii) A0 is dense in A[ · ]; (iii) for every x ∈ A0 , the map Rx : a ∈ A[ · ] → ax ∈ A[ · ] is continuous. A normed quasi *-algebra will be denoted as (A[ · ], A0 ). If A[ · ] is a Banach space, we say that (A[ · ], A0 ) is a Banach quasi *-algebra. The continuity of the involution implies that (iv) for every x ∈ A0 , the map Lx : a ∈ A[ · ] → xa ∈ A[ · ] is continuous too. If (A, A0 ) has no unit, it can always be embedded in a normed quasi *-algebra with unit e, using the standard procedure of unitization. In what follows, we shall always assume that (a) if ax = 0, for every x ∈ A0 , then a = 0; (b) if ax = 0, for every a ∈ A, then x = 0. Of course, both these conditions are automatically true if (A, A0 ) has a unit e. © Springer Nature Switzerland AG 2020 M. Fragoulopoulou, C. Trapani, Locally Convex Quasi *-Algebras and their Representations, Lecture Notes in Mathematics 2257, https://doi.org/10.1007/978-3-030-37705-2_3

31

32

3 Normed Quasi *-Algebras: Basic Theory and Examples

If (A[ · ], A0 ) is a normed quasi *-algebra, a norm topology can be defined on A0 in the following way. For x ∈ A0 , the following functions xL := sup ax a≤1

and

xR := sup xa, a≤1

x ∈ A0 , a ∈ A,

(3.1.1)

are well defined norms on A0 . It is easy to see that xL = x ∗ R (and, of course, xR = x ∗ L ), for every x ∈ A0 . Moreover, by (3.1.1) it follows that ax ≤ axL

and

xa ≤ axR ,

∀ a ∈ A, x ∈ A0 .

(3.1.2)

Again by (3.1.1) and together with (3.1.2), we deduce that xyL ≤ xL yL

and

xyR ≤ xR yR ,

∀ x, y ∈ A0 .

(3.1.3)

Remark 3.1.2 1. It is clear from (3.1.3) that A0 becomes a normed algebra under the preceding norms  · L and  · R , respectively. Moreover, from (3.1.2) we conclude that the left module multiplication in A is continuous, when A0 is endowed with  · R , while the right module multiplication in A is continuous, when A0 is endowed with  · L . In other words, A[ · ] is a left normed module over the normed algebra A0 [ · R ] and a right normed module over the normed algebra A0 [ · L ]. 2. Suppose that (A[ · ], A0 ) is a normed quasi *-algebra with unit e. Then, conditions (iii) and (iv) of Definition 3.1.1 are equivalent with the following condition: (v) the normed space A[ · ] is a normed module over the normed *-algebra A0 [ · 0 ], where   x0 := max xL , xR ,

∀ x ∈ A0 .

(3.1.4)

Indeed: first note that by setting a = e in (3.1.2), we see that the norms  · ,  · 0 are comparable on A0 [ · 0 ]. Moreover, from (3.1.3) the vector norm  · 0 is submultiplicative, therefore A0 [ · 0 ] is a normed *-algebra. Suppose now that (v) holds. Fix x in A0 . Then, the composition of the following continuous maps A[ · ] → A0 [ · 0 ] × A[ · ] → A[ · ] : a → (x, a) → xa, is exactly the map Rx , so that the condition (iii) of Definition 3.1.1 holds. In the same way, it is proved that (iv) is also true. Conversely, suppose that (iii) and (iv) of Definition 3.1.1 are valid. From (3.1.2), (3.1.4) we readily obtain that the left and right module multiplications of the normed A0 [ · 0 ]-module A[ · ] are continuous; hence (v) follows.

3.1 Basic Definitions and Facts

33

Corollary 3.1.3 If (A[ · ], A0 ) is a normed quasi *-algebra, then A0 [ · 0 ] is a normed *-algebra and we have xy ≤ xy0 , yx ≤ xy0 , ∀ x, y ∈ A0 . The previous inequalities follow from the submultiplicativity of  · 0 and (3.1.2), (3.1.4). Remark 3.1.4 Note that having a unit e in (A[ · ], A0 ), we may always suppose, without loss of generality, that e = 1, since otherwise we can simply define the equivalent norm  ·  on A, with a := a/e, a ∈ A and have e = 1. If (A[ · ], A0 ) has no unit, then the norms  · ,  · 0 cannot be compared, in general (see Example 3.1.6). Definition 3.1.5 A Banach quasi *-algebra (A[ · ], A0 ) is called a BQ*-algebra if A0 [ · 0 ] is a Banach *-algebra and a proper CQ*-algebra if A0 [ · 0 ] is a C*algebra.

3.1.1 Examples Example 3.1.6 (Banach Function Spaces) Many Banach function spaces provide examples of Banach quasi *-algebras since they often contain a dense *-algebra of functions. For instance, if I = [0, 1] then (Lp (I ), C(I )), where C(I ) denotes the C*-algebra of all continuous functions on I and p ≥ 1, is a Banach quasi *algebra (more precisely, as we shall see in Chap. 5, a proper CQ*-algebra [50], if C(I ) is endowed with the usual supremum norm  · ∞ ; actually in this case, one has  · 0 =  · ∞ ). Similarly (Lp (R), Cc0 (R)) is a Banach quasi *-algebra without unit; Cc0 (R) stands here for the *-algebra of continuous functions on R with compact support. As pointed out in Remark 3.1.4, it is clear that  · p and  · ∞ do not compare on Cc0 (R). Other examples of Banach quasi *-algebras are easily found among Sobolev spaces, Besov spaces etc. (see e.g., [29]) Example 3.1.7 (Noncommutative Lp -Spaces) Let M be a von Neumann algebra and  a normal semifinite faithful trace [24, 25] on M. Then, the completion of the *-ideal   Jp := X ∈ M : (|X|p ) < ∞ , with respect to the norm Xp := (|X|p )1/p ,

X ∈ M,

34

3 Normed Quasi *-Algebras: Basic Theory and Examples

is usually denoted by Lp () [70, 76] and is a Banach space consisting of operators affiliated with M. Then, (Lp (), Jp ) is a Banach quasi *-algebra (without unit). If  is a finite trace then (Lp (), M) is a BQ*-algebra. We shall come back to this example in Sect. 5.6. Example 3.1.8 (Hilbert Algebras) A Hilbert algebra [19, Section 11.7] is a *algebra A0 , which is also a pre-Hilbert space with inner product ·|·, such that (i) The map y → xy, x, y ∈ A0 , is continuous with respect to the norm defined by the inner product. (ii) xy|z = y|x ∗ z, ∀ x, y, z ∈ A0 . (iii) x|y = y ∗ |x ∗ , ∀ x, y ∈ A0 . (iv) A20 is total in A0 . Let H denote the Hilbert space, which is the completion of A0 with respect to the norm defined by the inner product. The involution of A0 extends to the whole of H, since (iii) implies that * is isometric. Then, (H, A0 ) is a Banach quasi *-algebra, which we name Hilbert quasi *-algebra. Remark 3.1.9 Let (A[ · ], A0 ) be a Banach quasi *-algebra and suppose that the norm  ·  of A is a Hilbertian norm; that is, it satisfies the parallelogram law a + b2 + a − b2 = 2a2 + 2b2 ,

∀ a, b ∈ A.

Then, as it is well known, an inner product and a norm can be introduced in A by 1 a + i k b2 , a = a|a1/2 , 4 3

a|b =

∀ a, b ∈ A.

k=0

It is easy to see that the conditions (i) and (iii) of Example 3.1.8 are satisfied on A0 and extend to A. Moreover, if (A[ · ], A0 ) has a unit, then (iv) is also trivially fulfilled. As for condition (ii), the operator Lx of left multiplication by x ∈ A0 is bounded, therefore it has an adjoint L∗x , but we do not know if L∗x = Lx ∗ . For this reason, we will call Hilbertian quasi *-algebra, a Banach quasi *-algebra (A[ · ], A0 ) such that the norm  ·  of A is a Hilbertian norm, just to distinguish this case from that considered in Example 3.1.8. A deeper analysis of Hilbertian quasi *-algebra will be performed in Sect. 5.5. Example 3.1.10 Let us consider again the situation described in Example 2.4.13. Let D× denote the Banach conjugate dual space of D[ · 1 ] endowed with the dual norm  · −1 , i.e., f −1 := sup |f (ξ )|, f ∈ D× . ξ 1 ≤1

The Hilbert space H is canonically identified with a subspace of D× , by the map ξ → fξ , where fξ (η) = ξ |η, for every η ∈ D. The form b(·, ·) that puts D and

3.1 Basic Definitions and Facts

35

D× in conjugate duality is an extension of the inner product of D, so that we adopt the same symbol for both. The space L(D, D× ) of all continuous linear maps from D[·1 ] into D× [·−1 ] (see Sect. 2.1.6) carries a natural involution X → X† , X ∈ L(D, D× ) and a norm  · L , defined by ξ |X† η = η|Xξ , XL := sup Xξ −1 , ξ, η ∈ D, X ∈ L(D, D× ). ξ 1 ≤1

The involution X → X† is isometric and since D× [ · −1 ] is complete, L(D, D× )[ · L ] is a Banach space. The space B(D, D) (see also Example 2.4.13) has a natural norm  · B too defined by   ϕB := sup |ϕ(ξ, η)| : ξ 1 = η1 = 1 , ϕ ∈ B(D, D). With this norm, B(D, D) is a Banach space. Moreover, ϕ ∗ B = ϕB , for every ϕ ∈ B(D, D). In particular, for each ϕ ∈ B(D, D), there exists X ∈ L(D, D× ), such that ϕ := ϕX , where ϕX (ξ, η) = ξ |Xη,

ξ, η ∈ D.

It is easy to prove, in this case, that ϕB = XL . If D[·1 ] is a Banach space, then the converse is also true, i.e., if X ∈ L(D, D× ), then ϕX ∈ B(D, D) and the map X → ϕX is an isometric *-isomorphism [2, Ch. 10]. Let M be an O*-algebra on D, (in the sense that M is a *-subalgebra of L† (D)), with the property that each X ∈ M is continuous from D[ · 1 ] into itself (this is always true if D[ · 1 ] is a reflexive space). Then, (M, M), where M denotes the closure of M in L(D, D× )[ · L ], is a Banach quasi *-algebra. For instance, let us assume that D = D(S), where S is a positive selfadjoint operator with domain D(S), dense in H. If S ≥ I, then D is a Hilbert space with norm  · S defined by ξ S = Sξ , ξ ∈ D.  From now on, the symbol “” will denote a topological isomorphism.

From the above we have, L(D, D× )  B(D, D) and L† (D) = L† (D). If A ∈ then

L† (D),

  ϕA B = sup |Aξ |η| : Sξ  = Sη = 1 = S −1 AS −1 .

(3.1.5)

For every O*-algebra M on D, (M, M) is a Banach quasi *-algebra. Now we check that M is not a *-algebra, in general. From the above discussion it follows that the set S −1 MS −1 is a *-invariant vector space of bounded operators on H. We denote by MS its norm closure in B(H). Let   M†S := ϕ ∈ B† (D) : ϕ = ϕA , A ∈ M

36

3 Normed Quasi *-Algebras: Basic Theory and Examples

and M†S its closure in B(D, D)[ · B ]. Then,   M†S ⊆ ϕ ∈ B(D, D) : ϕ(ξ, η) = Y Sξ |Sη, ∀ ξ, η ∈ D and some Y ∈ MS . Indeed, if ϕ ∈ M†S , then there exists a sequence {ϕn } ⊂ M†S converging to ϕ. Clearly, ϕn = ϕAn , An ∈ L† (D). Since the sequence {ϕn } is Cauchy, (3.1.5) yields that S −1 (An − Am )S −1  → 0. Hence, S −1 An S −1 → Y , for some Y ∈ B(H). Clearly, Y ∈ MS and sup

Sξ =Sη=1

|ϕ(ξ, η) − Y Sξ |Sη| ≤

sup

Sξ =Sη=1

+

sup

|ϕ(ξ, η) − An ξ |η|

Sξ =Sη=1

|An ξ |η − Y Sξ |Sη| → 0,

since sup

Sξ =Sη=1

|An ξ |η − Y Sξ |Sη| = =

sup

ξ  =η =1

|An S −1 ξ  |S −1 η  − Y ξ  |η |

S −1 An S −1 − Y .

On the other hand, it is easily seen that M†S  M. Hence, if X ∈ M, then for some Y in MS , we have Xξ |η = Y Sξ |Sη,

∀ ξ, η ∈ D.

Thus, if Y S(D) is not a subset of D, then X is neither an element of M nor an operator on the Hilbert space H, but a true element of L(D, D× ).

3.1.2 Auxiliary Seminorms Now we come to the main topic of this section. We shall define some seminorms (one of them is, in fact, an unbounded C*-seminorm), closely related with families of sesquilinear forms [77, 78, 85] and examine their interplay with the family of *representations of a given quasi *-algebra (A, A0 ). In the case, where (A[ · ], A0 ) is a normed quasi *-algebra, some information on the structure of (A[ · ], A0 ) can be obtained by means of them. To begin with, let us fix some terminology. Let (A, A0 ) be a quasi *-algebra and s a seminorm on A. A sesquilinear form ϕ on A × A is said to be s-bounded if there exists a positive constant γ , such that |ϕ(a, b)| ≤ γ s(a)s(b),

∀ a, b ∈ A.

3.1 Basic Definitions and Facts

37

In this case, we put ϕs :=

sup

s(a)=s(b)=1

|ϕ(a, b)| = sup ϕ(a, a). s(a)=1

If s is exactly the norm of A, we say bounded instead of s-bounded and we write ϕ instead of ϕs . Let us now define   qI (a) := sup ϕ(ax, ax)1/2 : ϕ ∈ IA0 (A), x ∈ A0 , ϕ(x, x) = 1

(3.1.6)

and   D(qI ) := a ∈ A : qI (a) < ∞ . Remark 3.1.11 If (A, A0 ) has a unit e, then one can easily check that   qI (a) = sup ϕ(a, a)1/2 : ϕ ∈ IA0 (A), ϕ(e, e) = 1 . Proposition 3.1.12 Let (A, A0 ) be a quasi *-algebra. For each ϕ ∈ IA0 (A), let πϕ denote the corresponding GNS representation. Then,  D(qI ) = a ∈ A : πϕ (a) ∈ B(Hϕ ), ∀ ϕ ∈ IA0 (A) and 

= a ∈ A : π(a) isbounded, ∀ π ∈ Repr (A) and

sup

πϕ (a) < ∞

sup

π(a) < ∞

ϕ∈IA0 (A)

π ∈Repr (A)

and qI (a) =

sup

ϕ∈IA0 (A)

πϕ (a) =

sup

π ∈Repr (A)

π(a),

∀ a ∈ D(q).

(3.1.7)

Proof Without loss of generality, we may assume that (A, A0 ) has a unit e. For shortness we put  A1 = a ∈ A : πϕ (a) ∈ B(Hϕ ), ∀ ϕ ∈ IA0 (A) and

sup

ϕ∈IA0 (A)

πϕ (a) < ∞

and  A2 = a ∈ A : π(a) is bounded, ∀ π ∈ Repr (A) and

sup

π ∈Repr (A)

π(a) < ∞ .

38

3 Normed Quasi *-Algebras: Basic Theory and Examples

Let a ∈ D(qI ). Then, if ϕ ∈ IA0 (A), we have ϕ(ax, ax) ≤ qI (a)2 ϕ(x, x),

∀ x ∈ A0 .

Hence, πϕ (a) is bounded and πϕ (a) ≤ qI (a). Therefore, a ∈ A1 and sup

ϕ∈IA0 (A)

πϕ (a) ≤ qI (a).

Let a ∈ A1 . Clearly, πϕ (a) is bounded, if and only if, πϕ◦ (a) is bounded. Since πϕ◦ is regular (Proposition 2.4.16), we conclude that sup

ϕ∈IA0 (A)

πϕ (a) ≤

sup

π ∈Repr (A)

π(a).

(3.1.8)

On the other hand, if π ∈ Repr (A) then, for every ξ ∈ Dπ , we consider the corresponding vector form ϕξ . The regularity of π implies that ϕξ ∈ IA0 (A) (see discussion after (2.4.12)) and so the *-representation πϕ◦ξ , with cyclic vector ξϕ = λϕξ (e), can be constructed. By assumption, the operator πϕ◦ξ (a) is bounded. Then, we have π(a)ξϕ 2 = πϕ◦ξ (a)λϕξ (e)2 ≤ πϕ◦ξ (a)2 ξϕ 2 , which implies that π(a) is bounded and that the converse inequality in (3.1.8) holds; i.e., A1 ⊆ A2 . Therefore, it is sufficient to prove the equalities D(qI ) = A2 and qI (a) =

sup

π ∈Repr (A)

π(a),

∀ a ∈ D(qI ).

Now, let π ∈ Repr (A) and, for ξ ∈ Dπ , define ϕξ as in (2.4.12). From (3.1.6) it follows that ϕξ (ax, ax) ≤ qI (a)2 ϕξ (x, x),

∀ a ∈ D(qI ), x ∈ A0 .

This implies that π(a)ξ 2 = ϕξ (a, a) ≤ qI (a)2 ϕξ (e, e) = qI (a)2 ξ 2 ,

∀ a ∈ D(qI ).

Thus, for every a ∈ D(qI ), π(a) is a bounded operator and sup

π ∈Repr (A)

π(a) ≤ qI (a) < ∞.

Conversely, if π(a) is bounded, for every π ∈ Repr (A), and supπ ∈Repr (A) π(a) < ∞, this is in particular true for the representation πϕ◦ constructed by any ϕ ∈

3.1 Basic Definitions and Facts

39

IA0 (A). Thus, ϕ(ax, ax) = πϕ◦ (a)λϕ (x)2 ≤ =

 sup

π ∈Repr (A)

 sup

π ∈Repr (A)

 π(a)2 λϕ (x)2

 π(a)2 ϕ(x, x),

∀ a ∈ A, x ∈ A0 .

Therefore a ∈ D(qI ) and qI (a) ≤

sup

π ∈Repr (A)

π(a).  

This completes the proof.

Lemma 3.1.13 Let (A[ · ], A0 ) be a normed quasi *-algebra. Set (see also Definition 2.3.1)   PA0 (A) := ϕ ∈ QA0 (A) : ϕ is bounded . Then, PA0 (A) ⊆ IA0 (A). Proof If ϕ ∈ PA0 (A), then the subspace λϕ (A0 ) is dense in Hϕ . Indeed, if a ∈ A, there exists a sequence {xn }, xn ∈ A0 , such that xn → a in A. Then, we have n→∞

λϕ (a) − λϕ (xn )2 = ϕ(a − xn , a − xn ) ≤ ϕ2 a − xn 2 → 0. n→∞

 

Definition 3.1.14 We define   SA0 (A) := ϕ ∈ PA0 (A) : ϕ ≤ 1 . More explicitly, ϕ ∈ SA0 (A), if and only if, it satisfies the following conditions: (i) ϕ(a, a) ≥ 0, ∀ a ∈ A; (ii) ϕ(ax, y) = ϕ(x, a ∗ y), ∀ a ∈ A, x, y ∈ A0 ; (iii) |ϕ(a, b)| ≤ a b, ∀ a, b ∈ A. Remark 3.1.15 Of course, the possibility that SA0 (A) = {0} is not excluded and examples, where this happens, can be given (see, e.g., Example 3.1.29 below). If (A[ · ], A0 ) is a Banach quasi *-algebra, we denote the Banach dual space of A by A . Then, A can be made into a Banach A0 -bimodule under the norm f  := sup |f (x)|, f ∈ A , x≤1

40

3 Normed Quasi *-Algebras: Basic Theory and Examples

by defining, for f ∈ A , x ∈ A0 , the module operations (f, x) → f ◦ x (resp. (x, f ) → x ◦ f ), in the following way: (f ◦ x)(a) := f (ax), (x ◦ f )(a) := f (xa), a ∈ A. As usual, an involution f → f ∗ can be defined on A by f ∗ (a) := f (a ∗ ), a ∈ A. Under this notation we can easily prove the following (see, also [86]): Proposition 3.1.16 Let (A[ · ], A0 ) be a Banach quasi *-algebra and ϕ a positive sesquilinear form on A × A. The following statements are equivalent: (i) ϕ ∈ SA0 (A); (ii) there exists a bounded conjugate linear operator T : A → A with the properties: (ii.1) (T a)(a) ≥ 0, ∀ a ∈ A; (ii.2) T (xa) = (T a) ◦ x ∗ , ∀ x ∈ A0 , a ∈ A, where ◦ as above; (ii.3) T B(A,A ) ≤ 1, B(A, A ) standing for the algebra of all bounded conjugate linear operators from A in A ; (ii.4) ϕ(a, b) = (T b)(a), ∀ a, b ∈ A.

3.1.3 Sufficient Family of Forms and *-Semisimplicity Definition 3.1.17 Let (A[ · ], A0 ) be a Banach quasi *-algebra. We say that SA0 (A) (see discussion before Remark 3.1.15) is sufficient, if a ∈ A with ϕ(a, a) = 0, for each ϕ ∈ SA0 (A), implies that a = 0. Lemma 3.1.19, just below, allows us to formulate in various ways the notion of sufficiency for SA0 (A). Before we state it, we give the definition of a seminorm p, needed in this lemma, but also in the sequel. Let (A[ · ], A0 ) be a normed quasi *-algebra. We put p(a) :=

sup

ϕ∈SA0 (A)

ϕ(a, a)1/2 , a ∈ A.

(3.1.9)

Then, p is a seminorm on A, with p(a) ≤ a, for every a ∈ A (see Definition 3.1.14(iii)). Proposition 3.1.18 Let (A[ · ], A0 ) be a Banach quasi *-algebra. Then, for every a ∈ A, there exists ϕ ∈ SA0 (A), such that ϕ(a, a) = p(a). Proof Let a ∈ A be fixed. By the definition of p itself, it follows that if ε > 0 and n ∈ N, there exists ϕε,n ∈ SA0 (A), such that p(a)2 − 2εn < ϕε,n (a, a). We define ∞ 

1 ϕ(a , b ) := ϕε,n a  , b , 2n 



n=1

∀ a  , b ∈ A.

3.1 Basic Definitions and Facts

41

It is easy to verify that ϕ ∈ SA0 (A); in particular ∞  ∞  1        1   ≤ ϕε,n (a  , b ) a ϕ , b |ϕ a , b | =   ε,n n n   2 2 n=1 n=1 ∞   1 ≤ a  b  = a  b , ∀ a  , b ∈ A. 2n n=1

Moreover, ϕ(a, a) =

∞ ∞ 

 1 1 ε 4 a, a > p(a)2 − n = p(a)2 − ε. ϕ ε,n n n 2 2 2 3 n=1

n=1

Then, since ε is arbitrary, ϕ( a, a) ≥ p(a)2 . But also ϕ(b, b) ≤ p(b)2 , for every   b ∈ A. Thus, we conclude that ϕ(a, a) = p(a)2 . Lemma 3.1.19 Let (A[ · ], A0 ) be a Banach quasi *-algebra with unit e. For an element a ∈ A, the following statements are equivalent: (i) (ii) (iii) (iv) (v) (vi)

p(a) = 0; ϕ(a, a) = 0, ∀ ϕ ∈ SA0 (A); ϕ(a, b) = 0, ∀ ϕ ∈ SA0 (A) and b ∈ A; ωϕ (a) = 0, ∀ ϕ ∈ SA0 (A); ϕ(ax, x) = 0, ∀ ϕ ∈ SA0 (A) and x ∈ A0 ; ϕ(ax, y) = 0, ∀ ϕ ∈ SA0 (A) and x, y ∈ A0 .

Definition 3.1.20 Let (A[ · ], A0 ) be a normed quasi *-algebra with unit e and π a *-representation of (A[ · ], A0 ) into L† (Dπ , Hπ ) (see Definition 2.2.5 and discussion before Sect. 2.2). We say that π is • weakly continuous, if π is continuous from A[ · ] into L† (Dπ , Hπ )[tw ]; • strongly continuous, if π is continuous from A[ · ] into L† (Dπ , Hπ )[ts ]; • strongly*continuous, if π is continuous from A[ · ] into L† (Dπ , Hπ )[ts ∗]. It is easy to prove that a *-representation is strongly continuous, if and only if, it is strongly* continuous. We generalize now some familiar notions from Banach *algebras theory, such as that of *-radical and of *-semisimplicity; for the Banach case, see Definition A.5.2. Definition 3.1.21 If (A[ · ], A0 ) is a normed quasi *-algebra, we define the *radical RA0 (A) of (A[·], A0 ) as the intersection of the kernels of all of its strongly continuous qu*-representations (see Sect. 2.2). Proposition 3.1.22 Let (A[ · ], A0 ) be a normed quasi *-algebra with unit e. For an element a ∈ A the following statements are equivalent: (i) a ∈ RA0 (A); (ii) ϕ(a, a) = 0, for every ϕ ∈ SA0 (A).

42

3 Normed Quasi *-Algebras: Basic Theory and Examples

Proof Assume that (i) holds and let ϕ ∈ SA0 (A). Let (πϕ , λϕ , Hϕ ) be the GNS construction of Proposition 2.4.1. Then, the restriction πϕ◦ of πϕ to λϕ (A0 ) is a qu*representation. Since, πϕ◦ (b)λϕ (x)2 = ϕ(bx, bx) ≤ bx2 ≤ b2 x2 ,

∀ b ∈ A, x ∈ A0 ,

πϕ◦ is strongly continuous. Hence, πϕ◦ (a) = 0. Then, ϕ(a, a) = πϕ◦ (a)λϕ (e)2 = 0. Conversely, assume that (ii) holds. Let π be any strongly continuous qu*-representation of (A[ · ], A0 ). Then, there exists γπ > 0, such that π(a)ξ  ≤ γπ a,

∀ a ∈ A.

Let ξ ∈ Dπ , with ξ  = 1. Define ϕπ (a, b) := π(a)ξ |π(b)ξ , a, b ∈ Dπ . Then, ϕπ (a, b) = |π(a)ξ |π(b)ξ | ≤ π(a)ξ  π(b)ξ  ≤ γπ2 ab,

∀ a, b ∈ Dπ .

It is easy to prove that ϕπ /γπ2 ∈ SA0 (A). Thus, ϕπ (a, a) = 0 and this implies that π(a)ξ  = 0. Since, ξ is arbitrary, we conclude that π(a) = 0.   Definition 3.1.23 A normed quasi *-algebra (A[ · ], A0 ) is called *-semisimple if RA0 (A) = {0}. An immediate consequence of the previous Proposition 3.1.22 is the following Corollary 3.1.24 A normed quasi *-algebra (A[ · ], A0 ) is *-semisimple, if and only if, SA0 (A) is sufficient. Let (A[ · ], A0 ) be a Banach quasi *-algebra. We denote by A+ 0 the set of positive elements of A0 ; i.e.,  A+ 0 :=

n 

 xk∗ xk : xk ∈ A0 , k = 1, . . . , n, n ∈ N .

k=1 ·

, the closure of A+ We put A+ := A+ 0 0 in the norm topology of A. Elements of A+ are called positive too and we often write a ≥ 0 instead of a ∈ A+ . If a ∈ A+ , then a = a ∗ , as it follows immediately by the definition. Moreover, A+ is a convex cone.

3.1 Basic Definitions and Facts

43

A linear functional ω on A is called positive if ω(a) ≥ 0, for every a ∈ A+ . We have already denoted by A the Banach dual space of A[·]. Then, the set of positive elements of A (that is, the set of bounded positive functionals) is denoted by A+ . In this regard, we have the following Proposition 3.1.25 Let (A[ · ], A0 ) be a Banach quasi *-algebra with unit e. If the linear span of the set   AP := ωϕ : ϕ ∈ PA0 (A) is weakly*-dense in A , then SA0 (A) is sufficient or, equivalently, (A[ · ], A0 ) is *-semisimple. Conversely, if A[ · ] is a reflexive Banach space and SA0 (A) is sufficient, then the linear span of AP is weakly*-dense in A . Proof Assume that SA0 (A) is not sufficient. Then, there exists a ∈ A, a = 0, such that for every ϕ ∈ SA0 (A), ϕ(a, a) = 0. This implies that ωϕ (a) = 0, for each ϕ ∈ SA0 (A). Thus, the non-zero continuous linear functional fa on A , defined by fa (ω) := ω(a), is zero all over the set AP and therefore, on its linear span. Hence, this set is not weakly*-dense in A . Conversely, assume that the linear span of AP is not weakly*-dense in A . Then, by the reflexivity of A[ · ], there would exist an element a ∈ A, a = 0, such that ωϕ (a) = ϕ(a, e) = 0, for each ϕ ∈ PA0 (A) [28, p. 186, Corollary 1]. Thus, by Lemma 3.1.19, we obtain ϕ(a, a) = 0, for each ϕ ∈ AP . This implies a = 0, a   contradiction.  Here and in Appendix A, for simplicity’s sake, we shall denote the unit ball of

A by UA , instead of U (A ), which is our standard symbol for the unit ball of a normed space (see discussion before Proposition 4.1.28).    Let UA := ω ∈ A : ω ≤ 1 be the unit ball of A and AS = ωϕ :  ϕ ∈ SA0 (A) . Of course, it is truly possible that SA0 (A) = {0} (or, equivalently, AS = {0}). It is, clearly, much more interesting to consider Banach quasi *-algebras, for which the set SA0 (A) is sufficiently rich (Sect. 3.1.3).

Proposition 3.1.26 Assume that (A[ · ], A0 ) has a unit and that SA0 (A) = {0}. Then, the following statements hold: (i) AS is a convex, weakly*-compact subset of UA ; (ii) AS has extreme points. If ωϕ is extreme, then ϕ = 1; (iii) ωϕ is extreme in AS , if and only if, ϕ is extreme in SA0 (A). The proof is very simple and we omit it. Proposition 3.1.27 Let (A[ · ], A0 ) be a *-semisimple Banach quasi *-algebra with unit. Let a ∈ A. Then, the following hold: (i) a = a ∗ , if and only if, ωϕ (a) ∈ R, for each ϕ ∈ SA0 (A); (ii) if a ≥ 0, then ωϕ (a) ≥ 0, for each ϕ ∈ SA0 (A); (iii) a ∈ A+ ∩ {−A+ }, if and only if, a = 0.

44

3 Normed Quasi *-Algebras: Basic Theory and Examples

Proof (i) Assume that, for each ϕ ∈ SA0 (A), ωϕ (a) ∈ R. Then, we have ωϕ (a − a ∗ ) = ωϕ (a) − ωϕ (a ∗ ) = ωϕ (a) − ωϕ (a) = 0, for every ϕ ∈ SA0 (A). By Lemma 3.1.19 one has ϕ(a − a ∗ , a − a ∗ ) = 0, for every ϕ ∈ SA0 (A). Hence, a = a ∗ . The converse implication is obvious. Lemma 3.1.19 is also used in the proof of (ii) and (iii). (ii) If a ≥ 0, then by definition a is the limit of a sequence of elements of A+ 0 and every ωϕ is positive on A0 and continuous on A[ · ] (for the latter, see Definition 3.1.14). (iii) Assume that a ∈ A+ ∩ {−A+ }; then, by (ii) it follows that ωϕ (a) = 0, for every ϕ ∈ SA0 (A). From this we conclude that a = 0.   Proposition 3.1.28 Let (A[ · ], A0 ) be a *-semisimple BQ*-algebra. Then, A0 is a *-semisimple Banach *-algebra. Proof It suffices to show that if x ∈ A0 and ω(x ∗ x) = 0, for each positive linear functional ω on A0 , then x = 0 . If this assumption is satisfied then, in particular, we will have ωϕ (x ∗ x) = 0, for each ϕ ∈ SA0 (A). This implies that ϕ(x, x) = 0, for every ϕ ∈ SA0 (A) and thus x = 0.   The case of a *-semisimple normed quasi *-algebra (i.e., with trivial *-radical) is particularly interesting. The main reason is that, for a *-semisimple normed quasi *-algebra, it is possible to define a refinement of the partial multiplication. In this way, the lattices of multipliers become nontrivial. We will now sketch the construction that makes of any *-semisimple Banach quasi *-algebra a nontrivial partial *-algebra. But before going forth let us examine an example. Example 3.1.29 In this example we shall discuss the *-semisimplicity of the Banach quasi *-algebra (Lp (I ), C(I )), where I = [0, 1] and p ≥ 1 (Example 3.1.6). We consider here the spaces Lp with respect to the Lebesgue measure dt. However, it is not difficult to realize that the same argument holds if in the place of I is a compact Hausdorff space X with a Borel measure μ. The following two well-known facts will be needed: (lp.i) Let y be a measurable function on I , and assume that xy ∈ Lr (I ), for all x ∈ Lp (I ) with 1 ≤ r ≤ p. Then, y ∈ Lq (I ), with p−1 + q −1 = r −1 . (lp.ii) Let p, q, r ≥ 1, such that p−1 + q −1 = r −1 . Let w ∈ Lq (I ). Then, the linear operator Tw : x ∈ Lp (I ) → xw ∈ Lr (I ) is bounded and Tw p,r = wq , where Tw p,r denotes the norm of Tw as bounded operator from Lp (I ) into Lr (I ). We put   p B+ = v ∈ Lp/(p−2) (I ), v ≥ 0 and vp/(p−2) ≤ 1 . If p = 2, we set

p p−2

= ∞.

3.1 Basic Definitions and Facts

45 p

Statement 1 Let p ≥ 2. Then, ϕ ∈ SC(I ) (Lp (I )), if and only if, there exists v ∈ B+ , such that  (3.1.10) ϕ(x, y) = x(t)y(t)v(t) dt, ∀ x, y ∈ Lp (I ). I

The sufficiency is straightforward. As for the necessity, we first notice that any bounded sesquilinear form ϕ on Lp (I ) × Lp (I ) can be represented as  x(t)(T y)(t) dt, x, y ∈ Lp (I ),

ϕ(x, y) =

(3.1.11)

I 

where T is a bounded linear operator from Lp (I ) into its dual space Lp (I ), with p−1 + p−1 = 1 [17, Vol. II, §40]. From (lp.ii) and Eq. (3.1.11) it follows easily that T y = yT u,

∀ y ∈ Lp (I ),

where u(t) = 1, for each t ∈ I . Set v := T u; from (lp.i) we obtain v ≥ 0. By the positivity of ϕ, we get v ∈ Lp/(p−2) (I ). Making use of (lp.ii), it is also easy to check that vp/(p−2) ≤ 1. Statement 2 If 1 ≤ p < 2, then SC(I ) (Lp (I )) = {0}. Indeed, let v = 0. Then, we can choose α > 0, in such a way, that the set Iα = {t ∈ I : v(t) > α} has positive measure. Let x ∈ Lp (Iα ) \ L2 (Iα ) (such a function always exists because of the assumption on p). Now define   x (t) =

x(t), if t ∈ Iα 0, if t ∈ I \ Iα

Clearly,  x ∈ Lp (I ). Moreover, 



ϕ( x,  x) =



| x (t)|2 v(t)dt =

|x(t)|2 v(t)dt ≥ α

I

Iα

|x(t)|2 dt = ∞ Iα

and this is a contradiction. Statement 3 If p ≥ 2, then (Lp (I ), C(I )) is *-semisimple. We show first that for each x ∈ Lp (I ), there exists  ϕ ∈ SC(I ) (Lp (I )), such that 2  ϕ (x, x) = xp . This is achieved by setting  ϕ (y, z) =

2−p xp

 y z¯ |x|p−2 dt, y, z ∈ Lp (I ), I 2−p

where  ϕ ∈ SC(I ) (Lp (I )), since the function v = |x|p−2 xp p ϕ (x, x) = x2p . B+ . A direct calculation shows that 

belongs to the set

46

3 Normed Quasi *-Algebras: Basic Theory and Examples

Let us now suppose that ϕ(x, x) = 0, for all ϕ ∈ SC(I ) (Lp (I )). Then, in particular,  ϕ (x, x) = x2p = 0. Therefore, x = 0 and so (Lp (I ), C(I )) is *semisimple. In a *-semisimple Banach quasi *-algebra the multiplication can be refined (in the sense that we can define an extension of the partial multiplication) as follows (see also [39]). Definition 3.1.30 Let (A[ · ], A0 ) be a *-semisimple normed quasi *-algebra. We say that the weak multiplication, a b, a, b ∈ A, is well-defined if there exists c ∈ A, such that ϕ(bx, a ∗ y) = ϕ(cx, y),

∀ x, y ∈ A0 and ϕ ∈ SA0 (A).

In this case, we put a b := c. The following result is immediate. Proposition 3.1.31 Let (A[ · ], A0 ) be a *-semisimple normed quasi *-algebra. Then, A is also a partial *-algebra with respect to the weak multiplication. We shall denote by Rw (A) the space of universal right weak multipliers of A; i.e., the space of all b ∈ A, such that a b is well-defined, for every a ∈ A. In the same way Lw (A) is defined. Clearly, A0 ⊆ Rw (A) (resp. A0 ⊆ Lw (A)). Given a fixed a ∈ A, denote by Rw (a) the set of all b ∈ A, such that a b is well-defined. Similarly, we define Lw (a). Remark 3.1.32 The sesquilinear forms of SA0 (A) define on A, the topologies τw , τs , τs ∗ generated, respectively, by the following families of seminorms: τw : τs : τs ∗ :

a→  |ϕ(ax, y)|, a ∈ A, ϕ ∈ SA0 (A), x, y ∈ A0 ; a→  ϕ(a, a)1/2 , a ∈ A, ϕ ∈ SA0(A); a→  max ϕ(a, a)1/2 , ϕ(a ∗ , a ∗ )1/2 , a ∈ A, ϕ ∈ SA0 (A).

From continuity of ϕ ∈ SA0 (A), it follows that the topologies τw , τs , τs ∗ are coarser than the initial norm topology of A. Now we have the following Proposition 3.1.33 The following statements are equivalent: (i) the weak product a b is well-defined; (ii) there exists a sequence {yn } in A0 and c ∈ A, such that yn − b → 0 and τw ayn −→ c; (iii) there exists a sequence {xn } in A0 and c ∈ A, such that xn − a → 0 and τw xn b −→ c. Proof We prove only that (i) ⇔ (ii). The proof of (i) ⇔ (iii) is very similar. Assume that a b is defined. By the  · -density of A0 , there exists a sequence {yn } in A0

3.1 Basic Definitions and Facts

47

approximating b. Then, for every z, z ∈ A0

ϕ (ayn )z, z = ϕ(yn z, a ∗ z ) → ϕ(bz, a ∗ z ) = ϕ (a b)z, z , τw

i.e., ayn −→ a b. Conversely, assume the existence of a sequence {yn } in A0 τw approximating b such that ayn −→ c ∈ A. Then, for every z, z ∈ A0 , we have

ϕ(bz, a ∗ z ) = lim ϕ(yn z, a ∗ z ) = lim ϕ (ayn )z, z = ϕ(cz, z ), n→∞

n→∞

 

i.e., a b is well-defined.

Let (A[ · ], A0 ) be a Banach quasi *-algebra. To every a ∈ A we may correspond the linear maps La and Ra defined as follows x ∈ A0 → La x := ax ∈ A, x ∈ A0 → Ra x := xa ∈ A.

(3.1.12)

If (A[ · ], A0 ) is a *-semisimple Banach quasi *-algebra, then the weak multiplication  allows us to extend La , (resp., Ra ) to the set Rw (a) (resp., Lw (a)). a , (resp., R a ) these extensions. Then, L a b = a b, for every Let us denote by L a c = ca, for every c ∈ Lw (a). b ∈ Rw (a) and R Proposition 3.1.34 Let (A[ · ], A0 ) be a *-semisimple Banach quasi *-algebra. a , R a are closed linear maps in A[ · ]. Then, for every a ∈ A, L Proof We prove the statement only for La . Let bn − b → 0, with bn ∈ Rw (a) and a bn − c → 0. Then, for every ϕ ∈ SA0 (A) and for all z, z ∈ A0 ,



ϕ (a bn − c)z, z = ϕ (a bn )z, z − ϕ(cz, z ) = ϕ(bn z, a ∗ z ) − ϕ(cz, z ) → ϕ(bz, a ∗ z ) − ϕ(cz, z ) = 0. a By the *-semisimplicity, these relations show that b ∈ Rw (a) and c = a b; i.e., L   is closed. Example 3.1.35 We give here an example, where IA0 (A)  PA0 (A). We consider the Banach quasi *-algebra (L1 (I ), L∞ (I )), I = [0, 1]. For every x ∈ L1 (I ) we denote with x0 its restriction to Ia := [0, a], with 0 < a < 1. Define   A := x ∈ L1 (I ) : x0 ∈ L2 (Ia ) . Clearly (A, L∞ (I )) is a normed quasi *-algebra, when A is endowed with the norm induced by L1 (I ). It is easily shown that the positive sesquilinear form ϕ defined by  ϕ(x, y) =

a

x0 (t)y0 (t)dt, 0

x, y ∈ A,

48

3 Normed Quasi *-Algebras: Basic Theory and Examples

is an element of IA0 (A). In fact, in this case, A/Nϕ  L2 (I \ Ia ) and λϕ (L∞ (I ))  L∞ (I \ Ia ), which is dense in L2 (I \ Ia ). As shown in Example 3.1.29, SC(I ) (Lp (I )) = {0}; then, PC(I ) (L1 (I )) = {0} too. Therefore, ϕ ∈ PC(I ) (L1 (I )) = {0}. Example 3.1.36 Let us consider again a Banach quasi *-algebra of the type (M, M) constructed in Example 3.1.10. We put   D0 (M) = ξ ∈ D : Xξ ∈ H, ∀ X ∈ M . For ξ ∈ D0 (M), we define ϕξ (X, Y ) = Xξ |Y η,

X, Y ∈ M.

Then, it is easy to see that ϕξ ∈ QM (M). Following the definitions, it is easily seen that • ϕξ ∈ IM (M) ⇔ ξ ∈ D0 (M) and Xξ ∈ Mξ , ∀ X ∈ M, where Mξ denotes the closure of Mξ in H; • ϕξ ∈ PM (M) ⇔ ξ ∈ D0 (M) and sup Xξ  < ∞. XL ≤1

It is worth mentioning the fact that one can construct examples, where D0 (M) = {0}. For instance, let H = L2 (I ), with I = [0, 1] and D = Lp (I ), with p > 2. If η is a measurable function, denote by Mη the operator of multiplication by η. Consider as M the O*-algebra of multiplication operators by a function φ ∈ L∞ (I ), i.e.,   M = Mφ : φ ∈ L∞ (I ) . Then, it is easily seen that   M = Mφ : φ ∈ Lp/(p−2) (I ) and Mφ L = φp/(p−2) . Moreover, the following hold: • If 2 < p < 4, then D0 (M) = L2p/(4−p) (I ) and every ϕξ , ξ ∈ D0 (M), is bounded. • If p = 4, then D0 (M) = L∞ (I ) and again every ϕξ , ξ ∈ D0 (M), is bounded. • If p > 4, then D0 (M) = {0}. Lemma 3.1.37 below, will be often used in what follows. We remind that an m*-seminorm s on a *-algebra A0 is a seminorm satisfying the properties: (i) s(x ∗ ) = s(x), ∀ x ∈ A0 ; (ii) s(xy) ≤ s(x)s(y), ∀ x, y ∈ A0 .

3.1 Basic Definitions and Facts

49

Lemma 3.1.37 Let A0 be a *-algebra and ω a positive linear functional on A0 . Assume that there exists an m*-seminorm s on A0 , such that ∀ y ∈ A0 , ∃ γy > 0 : |ω(y ∗ xy)| ≤ γy s(x),

∀ x ∈ A0 .

Then, |ω(y ∗ xy)| ≤ s(x)ω(y ∗ y),

∀ x ∈ A0 .

Proof The argument for proof is based on Kaplansy’s inequality, using the assumption and applying repeatedly the Cauchy–Schwarz inequality.   Let us now define a new seminorm q as follows   q(a) := sup ϕ(ax, ax)1/2 : ϕ ∈ PA0 (A), x ∈ A0 , ϕ(x, x) = 1 ,

(3.1.13)

for all a ∈ A and   D(q) := a ∈ A : q(a) < ∞ . Clearly, D(qI ) ⊆ D(q) and q(a) ≤ qI (a), for every a ∈ D(qI ) (see (3.1.6)). As in the case of qI one can easily check that if (A, A0 ) has a unit e, then   q(a) = sup ϕ(a, a)1/2 : ϕ ∈ PA0 (A), ϕ(e, e) = 1 , a ∈ D(q). In order to obtain a description of D(q) similar to that of D(qI ), we give the following Definition 3.1.38 Let (A[ · ], A0 ) be a normed quasi *-algebra and π a *representation of (A[·], A0 ) with domain Dπ . We say that π is completely regular if, for every ξ ∈ Dπ , the positive sesquilinear form ϕξ is bounded. The set of all completely regular *-representations of (A[ · ], A0 ) is denoted by Repcr (A). Clearly, if π is completely regular, then it is regular. Proposition 3.1.39 Let (A[ · ], A0 ) be a normed quasi *-algebra. For each ϕ ∈ PA0 (A), let πϕ denote the corresponding GNS representation. Then,  D(q) = a ∈ A : πϕ (a) ∈ B(Hϕ ), ∀ ϕ ∈ PA0 (A) and sup

ϕ∈PA0 (A)



 πϕ (a) < ∞

 = a ∈ A : π(a) is bounded, ∀ π ∈ Repcr (A) and sup π(a) < ∞ π ∈Repcr (A)

and q(a) =

sup

ϕ∈PA0 (A)

πϕ (a) =

sup

π ∈Repcr (A)

π(a),

∀ a ∈ D(q).

(3.1.14)

50

3 Normed Quasi *-Algebras: Basic Theory and Examples

Proof The proof is very similar to that of Proposition 3.1.12, so we do not repeat all the details. The only point to be taken into account is that if ϕ ∈ PA0 (A), then the corresponding representation πϕ◦ is completely regular. Indeed, if ξ = λϕ (x), then (see (3.1.2), (3.1.4) and Definition 3.1.14(iii)) there exists γ > 0, such that ϕξ (a, a) = πϕ◦ (a)λϕ (x)|πϕ◦ (a)λϕ (x) = ϕ(ax, ax) ≤ γ x20 a2 , a ∈ A, x ∈ A0 .

 

Hence, ϕξ is bounded.

In what follows we will show that q plays, together with p, a crucial role for the structure of a normed or Banach quasi *-algebra. The following preliminary proposition has been given in [80]. The statement (i) below follows from Lemma 3.1.37, while (ii) can be easily deduced from Proposition 3.1.39. The statements (iii) and (iv) follow by the very definitions. Proposition 3.1.40 The following statements hold: (i) A0 ⊆ D(q) and q(x) ≤ x0 , ∀ x ∈ A0 ; (ii) q is an extended C*-seminorm on (A, A0 ) (i.e., q(a ∗ ) = q(a), ∀ a ∈ D(q) and q(x ∗ x) = q(x)2 , ∀ x ∈ A0 ; see [77]); (iii) p(ax) ≤ q(a)p(x), ∀ a ∈ D(q), x ∈ A0 ; (iv) p(xa) ≤ x0 p(a), ∀ a ∈ A, x ∈ A0 . Remark 3.1.41 If (A, A0 ) has a unit e, then from (iii) it follows that p(a) ≤ q(a), for every a ∈ D(q). Now, we put   N (p) := a ∈ A : p(a) = 0 . Then, by (iv) of Proposition 3.1.40 it follows that N (p) is a left qu-ideal of (A, A0 ). p We consider N0 (p) = N (p)∩A0 . Then, the quotient A0 := A0 /N0 (p) is a normed space with norm x +N0 (p)p := p(x), x ∈ A0 . Let us denote by Ap the completion p of A0 [ · p ]. Then, we have the following Proposition 3.1.42 The quotient A/N(p) can be identified with a dense subspace of Ap . Moreover, (i) if p(a) = p(a ∗ ), for every a ∈ A, then Ap is a Banach space with isometric p p involution extending the natural involution of A0 ; A0 is a *-algebra and p (Ap , A0 ) can be made into a Banach quasi *-algebra. (ii) If p(xa) ≤ p(x)p(a), for every x ∈ A0 and a ∈ A, then Ap is a Banach algebra. (iii) If p is an m*-seminorm on A0 , then Ap is a Banach *-algebra. (iv) If p is a C*-seminorm on A0 , then Ap is a C*-algebra. Proof Let a ∈ A. Then, there exists a sequence {xn } of elements of A0 , such that a − xn  → 0, as n → ∞. From the properties of p (see, after Definition 3.1.17) this implies that p(a − xn ) → 0, as n → ∞. We define  a :=  · p − limn→∞ (xn +

3.1 Basic Definitions and Facts

51

N0 (p)). By the construction of the completion,  a does not depend on the choice of the sequence {xn }. Moreover, the map j : a + N (p) ∈ A/N(p) →  a ∈ Ap is well-defined. Indeed, if a, a  ∈ A, such that a − a  ∈ N (p) and yn → a − a  with respect to the norm of A, then p(yn ) → 0 and so j (a − a  ) = 0, where j is injective. Indeed, assume that  a = 0 and let {xn } be a sequence of elements of A0 , such that a − xn  → 0, as n → ∞. Then,  · p − limn→∞ (xn + N0 (p)) = 0. Hence, p(xn ) → 0, as n → ∞. This, in turn, implies that p(a) = 0 and so a ∈ N (p). The proofs of (ii), (iii) and (iv) are easily checked. As for (i), from the definition of p and the assumptions it follows that N0 (p) is a *-ideal of A0 , therefore A0 /N0 (p) p is a *-algebra. For defining the multiplication that makes of (Ap , A0 ) a quasi *algebra, we proceed as follows: if z ∈ Ap , then z =  · p − limn→∞ (xn + N0 (p)), with xn ∈ A0 ; so taking also x ∈ A0 and using (iv) of Proposition 3.1.40, we obtain that the sequence (xxn + N0 (p)) is  · p -Cauchy. Thus, we can define xz =  · p − limn→∞ (xxn + N0 (p)).   Remark 3.1.43 We shall show below that (iii) and (iv) are indeed equivalent. Example 3.1.44 Let us consider the Banach quasi *-algebra (Lp (I ), C(I )) with I = [0, 1]. In this case, one has (see Example 3.1.29) that PC(I ) (Lp (I )) =

⎧ ⎨ {ϕw : w ∈ Lp/(p−2) (I ), w ≥ 0}, if p ≥ 2 ⎩

{0},

if 1 ≤ p < 2,

where  ϕw (x, y) =

x(t)y(t)w(t)dt,

x, y ∈ Lp (I ).

I

If 1 ≤ p < 2 both p and q are identically zero. If p ≥ 2, then one can prove  that p(x) = xp and q(x) = sup ϕw (x, x)1/2 : w ∈ Lp/(p−2) (I ), w1 ≤ 1 , which is finite, if and only if, x ∈ L∞ (I ). In fact, we obtain that q(x) = x∞ ,

∀ x ∈ L∞ (I ).

Example 3.1.45 Let A0 be a Hilbert algebra (Example 3.1.8) and H its Hilbert space completion. As we have seen (H, A0 ) is a Banach quasi *-algebra. Since the inner product of H is an element of PA0 (H), one has p(a) = a, for every a ∈ H. As for q it is easily seen that     D(q) = a ∈ H : La is bounded = a ∈ H : Ra is bounded

52

3 Normed Quasi *-Algebras: Basic Theory and Examples

and q(a) = La  = Ra ,

∀ a ∈ D(q),

where La : x ∈ A0 → ax ∈ H and Ra : x ∈ A0 → xa ∈ H. Thus, D(q) coincides, as already shown in [80], with the set of bounded elements of H. We conclude this section by discussing, in the case that (A[ · ], A0 ) is a Banach quasi *-algebra, some results on the automatic continuity for the classes of positive linear functionals and positive sesquilinear forms introduced so far. For this we need some lemmas (Lemmas 3.1.47 and 3.1.48), that are already known in slight different situations. For the sake of completeness we give the proofs adapted to the cases under consideration. We first remind the reader the definition of closed and closable sesquilinear form. Definition 3.1.46 Let B[ · ] be a Banach space and ϕ a positive sesquilinear form defined on D(ϕ) × D(ϕ), where D(ϕ) is a dense subspace of B[ · ]. Then, ϕ is said to be closed in D(ϕ) if D(ϕ) is complete under the norm  · ϕ defined by

1/2 aϕ := a2 + ϕ(a, a) , a ∈ D(ϕ). A form ϕ, with domain D(ϕ) is said to be closable if it has a closed extension ϕ to a domain D(ϕ) ⊆ B. In other words, ϕ is closed if, whenever a sequence {an } of elements of D(ϕ) converges to a ∈ B and ϕ(an − am , an − am ) → 0, as n, m → ∞, one has a ∈ D(ϕ) and ϕ(an − a, an − a) → 0, as n → ∞. The following lemma characterizes closed and everywhere defined forms. Lemma 3.1.47 Let B[ · ] be a Banach space and ϕ a positive sesquilinear form on B × B. The following statements are equivalent: (i) ϕ is lower semicontinuous; i.e., if {an } ⊂ B is a sequence converging to a ∈ B, with respect to  · , one has ϕ(a, a) ≤ lim infn→∞ ϕ(an , an ); (ii) ϕ is closed; (iii) ϕ is bounded. Proof (i) ⇒ (ii) Let {an } be a Cauchy sequence in B, with respect to  · ϕ . Then, for every ε > 0, there exists nε ∈ N, such that an − am 2 + ϕ(an − am , an − am ) < ε2 ,

∀ n, m > nε .

3.1 Basic Definitions and Facts

53

The completeness of B[ · ] implies the existence of an element a ∈ A, such that limn→∞ a − an  = 0. Now, fix m > nε and let n → ∞. We obtain a − am 2 + ϕ(a − am , a − am ) ≤ a − am 2 + lim inf ϕ(an − am , an − am ) ≤ ε2 . n→∞

Hence, a − am ϕ → 0. This proves that B is complete with respect to  · ϕ . (ii) ⇒ (iii) Since B[ · ] is a Banach space and a ≤ aϕ , for every a ∈ B, it follows that  ·  and  · ϕ are equivalent (by the inverse mapping theorem) and therefore ϕ is bounded. (iii) ⇒ (i) is straightforward.   For the terminology and notation applied just below, see discussion before Proposition 3.1.25. Lemma 3.1.48 Let (A[·], A0 ) be a Banach quasi *-algebra. Then, every positive linear functional ω on A is bounded on positive elements; i.e., there exists γ > 0, such that ω(a) ≤ γ a,

∀ a ∈ A+ .

Proof Suppose that ω is not bounded on A+ . Then, there would exist a sequence {an } of positive elements of A, such that an  ≤ 2−n and ω(an ) → ∞. Let b =  ∞ k=1 ak . Then, ω(b) = ω

∞  k=1

 ak

≥ω

 n  k=1

 ak

=

n 

ω(ak ) → ∞,

k=1

 

a contradiction.

Theorem 3.1.49 Let (A[ · ], A0 ) be a Banach quasi *-algebra satisfying the following condition: (D) every a = a ∗ ∈ A can be uniquely decomposed as a = a+ −a− , with a+ , a− ∈ A+ and a = a+  + a− . Then, every ϕ ∈ IA0 (A), such that ωϕx is positive, for every x ∈ A0 , is bounded. Proof Let x ∈ A0 . Since ωϕx , defined as in (2.4.14), is positive, Lemma 3.1.48 implies that ωϕx is bounded on positive elements; i.e., there exists γ > 0, such that ωϕx (a) ≤ γ a,

∀ a ∈ A+ .

Condition (D) then yields that, for every a = a ∗ ∈ A, |ωϕx (a)| = |ωϕx (a+ − a− )| ≤ ωϕx (a+ ) + ωϕx (a− ) ≤ γ (a+  + a− ) = γ a.

54

3 Normed Quasi *-Algebras: Basic Theory and Examples

The general statement is easily obtained by decomposing every z ∈ A as z = a +ib, with a = a ∗ , b = b∗ . Using the polarization identity, one proves easily that, for every x, y ∈ A0 , the linear functional Lx,y (a) := ϕ(ax, y), a ∈ A, is bounded. Let now {an } be a sequence in A and a ∈ A with limn→∞ an − a = 0. For every y ∈ A0 , by the Cauchy–Schwarz inequality, we have |ϕ(an , y)| ≤ ϕ(an , an )1/2 ϕ(y, y)1/2 . Taking the lim inf in both sides, we obtain |ϕ(a, y)| ≤ lim inf ϕ(an , an )1/2 ϕ(y, y)1/2 . n→∞

Now, since ϕ ∈ IA0 (A) (see Proposition 2.3.2 and discussion after it), there exists a sequence {xk } of elements of A0 such that ϕ(a − xk , a − xk ) → 0. This implies that lim ϕ(a, xk ) = ϕ(a, a)

k→∞

and

lim ϕ(xk , xk ) = ϕ(a, a).

k→∞

Then, from |ϕ(a, xk )| ≤ lim inf ϕ(an , an )1/2 ϕ(xk , xk )1/2 , n→∞

for k → ∞, we conclude that  1/2 ϕ(a, a) ≤ lim inf ϕ(an , an ) ϕ(a, a)1/2 . n→∞

Hence, ϕ(a, a) ≤ lim inf ϕ(an , an ); n→∞

 i.e., ϕ is lower semicontinuous. The statement then follows from Lemma 3.1.47. 

3.2 Continuity of Representable Linear Functionals As we have seen in Chap. 2 another way to construct a GNS-like representation of a quasi *-algebra (A, A0 ) is provided by representable linear functionals (Definition 2.4.6). Of course, for a normed or Banach quasi *-algebra, additional properties of the family R(A, A0 ) can be given and it is, moreover, of true interest considering the subclass Rc (A, A0 ) of continuous (or, equivalently, bounded) representable linear functionals on (A[ · ], A0 ). In particular, the question as to whether every representable linear functional is continuous deserves an answer. Investigations on this subject have been done in [31], to which we refer for more details.

3.2 Continuity of Representable Linear Functionals

55

Theorem 3.2.1 Let (A[ · ], A0 ) be a normed quasi *-algebra with unit e and ω a linear functional on A satisfying the conditions (L.1) and (L.2) of Definition 2.4.6. Then, the following statements are equivalent: (i) ω is representable; (ii) there exists a *-representation π defined on a dense domain Dπ of a Hilbert space Hπ and a vector ζ ∈ Dπ , such that ω(a) = π(a)ζ |ζ , for all a ∈ A; (iii) there exists a sesquilinear form  ∈ QA0 (A), such that ω(a) = (a, e), for all a ∈ A; (iv) there exists a Hilbert seminorm p on A, that is a seminorm satisfying the property p(a + b)2 + p(a − b)2 = 2p(a)2 + 2p(b)2 ,

∀ a, b ∈ A,

such that |ω(a ∗ x)| ≤ p(a)ω(x ∗ x)1/2 , for all x ∈ A0 . Proof (i) implies (ii) by Theorem 2.4.8. Suppose now that (ii) holds and define (a, b) := π(a)ζ |π(b)ζ , a, b ∈ A. It is then easy to check that  has the desired properties. This proves (iii). Now suppose that (iii) holds and define p(a) := (a, a)1/2 . Then, (iv) follows immediately from the Cauchy–Schwarz inequality. Finally suppose that (iv) holds. Then, 1

|ω(a ∗ x)| ≤ p(a)ω(x ∗ x) 2 ≤ γa ω(x ∗ x)1/2

1/2 where, for instance, γa ≡ 1 + p(a)2 . Hence, ω is representable.

 

To every ω ∈ R(A, A0 ), we can associate two sesquilinear forms, which are useful for our discussion. The first one  (already introduced in (iii) of Theorem 3.2.1), can be directly and equivalently computed by using the GNS representation πω , with cyclic vector ξω . Then, we can write the everywhere defined sesquilinear form ω as follows ω (a, b) := πω (a)ξω |πω (b)ξω , a, b ∈ A.

(3.2.15)

Thus, ω ∈ QA0 (A) and ω(a) = ω (a, e), for every a ∈ A. The second sesquilinear form, which we denote by ϕω is defined only on A0 ×A0 . To every ω ∈ R(A, A0 ) a sesquilinear form ϕω is associated, defined by ϕω (x, y) := ω(y ∗ x), x, y ∈ A0 . It is clear that ω extends ϕω . It is easy to see that (i) ϕω (x, x) ≥ 0, for every x ∈ A0 ; (ii) ϕω (xy, z) = ϕω (y, x ∗ z), for every x, y, z ∈ A0 .

(3.2.16)

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3 Normed Quasi *-Algebras: Basic Theory and Examples

Proposition 3.2.2 Let (A[ · ], A0 ) be a Banach quasi *-algebra, with unit e. For every ω ∈ Rc (A, A0 ), the positive sesquilinear form ϕω has a bounded extension ϕ ω to A × A and ϕ ω = ω . Proof Since ω is representable, there exists a Hilbert space Hω , a linear map λω : A0 → Hω and a *-representation πω with values in L† (λω (A0 ), Hω ) such that ω(y ∗ ax) = πω (a)λω (x)|λω (y),

∀ a ∈ A, x, y ∈ A0 .

By the continuity of ω, (3.1.4) and the discussion before (3.1.4), we obtain that for every a ∈ A and x, y ∈ A0 , |πω (a)λω (x)|λω (y)| = |ω(y ∗ ax)| ≤ γ ax0 y0 , γ > 0.

(3.2.17)

Now, consider the sesquilinear form ω defined in (3.2.15). As already noticed, ω extends ϕω . It remains to show that ω is closable. Suppose that {an } is a sequence in A, such that an  → 0 and ω (an − am , an − am ) = πω (an − am )ξω 2 → 0. Then, the sequence πω (an )ξω converges to a vector ζ ∈ Hω . Thus, πω (an )ξω |λω (y) → ζ |λω (y),

∀ y ∈ A0 .

Using (3.2.17) and the density of λω (A0 ) in Hω , we obtain ζ = 0. Hence, ω (an , an ) → 0; i.e., ω is closable (Definition 3.1.46). Thus, ω is closed and everywhere defined, hence bounded (Lemma 3.1.47). It is clear that ω = ϕ ω .   It appears natural for a normed quasi *-algebra to give a stronger notion of a representable linear functional by requiring a better control on the constant γa , which appears in the condition (L.3) of Definition 2.4.6. For this reason we will define uniformly representable linear functionals. In the Banach case, as we shall see this new notion is equivalent to the continuity of the representable linear functional under consideration. Definition 3.2.3 Let (A[ · ], A0 ) be a normed quasi *-algebra. We say that ω ∈ R(A, A0 ) is uniformly representable if there exists γ > 0, such that |ω(a ∗ x)| ≤ γ aω(x ∗ x)1/2 ,

∀ a ∈ A, x ∈ A0 .

(3.2.18)

The set of uniformly representable linear functionals is denoted by Ruc (A, A0 ). Proposition 3.2.4 Let (A[ · ], A0 ) be a Banach quasi *-algebra with unit e and ω ∈ R(A, A0 ). The following statements are equivalent: (i) ω ∈ Rc (A, A0 ); i.e., ω is bounded; (ii) ω ∈ Ruc (A, A0 ); i.e., ω is uniformly representable.

3.2 Continuity of Representable Linear Functionals

57

Proof (i) ⇒ (ii) Consider the operator Tω of Proposition 2.4.12, which is defined by Tω (a) := ξωa , for all a ∈ A. Since Tω is everywhere defined, it is enough to prove that Tω is closable and use the closed graph theorem. Let {an } be a sequence in A, such that an  → 0 and Tω an → ξ ∈ Hω . Then, from (2.4.9), we have λω (x)|ξ  = lim λω (x)|Tω an  = lim λω (x)|ξωan  = lim ω(an∗ x), n→∞

n→∞

n→∞

∀ x ∈ A0 .

But, for every x ∈ A0 , the continuity of ω and (3.1.4), give |ω(an∗ x)| ≤ can∗ x ≤ can x0 → 0, c > 0. Hence, λω (x)|ξ  = 0,

∀ x ∈ A0 .

Then, the density of λω (A0 ) in Hω implies that ξ = 0. Hence, Tω is closable. The closed graph theorem then implies that Tω is bounded. Furthermore, using again (2.4.9), we have |ω(a ∗ x)| = |λω (x)|Tω a| ≤ λω (x)Tω a ≤ Tω aω(x ∗ x)1/2 ,

∀ a ∈ A, x ∈ A0 .

That is ω ∈ Ruc (A, A0 ). (ii) ⇒ (i) is obvious.

 

We define a partial order in R(A, A0 ) as follows. If ω, θ ∈ R(A, A0 ) we say that ω ≤ θ if ω (a, a) ≤ θ (a, a), for every a ∈ A. By a slight modification of standard arguments (see, e.g., [2, Proposition 9.2.3]), one can prove the following Lemma 3.2.5 Let ω, ρ ∈ R(A, A0 ) and let πω denote the GNS representation associated to ω. If ρ ≤ ω, there exists an operator T ∈ (πω (A), Dω )w , with 0 ≤ T ≤ I , such that ρ (a, b) = πω (a)ξω |T πω (b)ξω ,

∀ a, b ∈ A.

Proof Using the Cauchy–Schwarz inequality, we get |ρ (a, b)|2 ≤ ρ (a, a)ρ (b, b) ≤ ω (a, a)ω (b, b) = πω (a)ξω 2 πω (b)ξω 2 ,

∀ a, b ∈ A.

Thus, the equality ρ (a, b) := πω (a)ξω |πω (b)ξω , a, b ∈ A, gives a densely defined, bounded sesquilinear form on A × A and there exists a unique bounded

58

3 Normed Quasi *-Algebras: Basic Theory and Examples

operator T in Hω [20, p. 44, Corollary], such that ρ (a, b) = πω (a)ξω |T πω (b)ξω ,

∀ a, b ∈ A.

The fact that ρ ≤ ω easily implies that 0 ≤ T ≤ I . Moreover, for every a ∈ A, x, y ∈ A0 , πω (a)πω (x)ξω |T πω (y)ξω  = ρ (ax, y) = ρ (x, a ∗ y) = πω (x)ξω |T πω (a ∗ )πω (y)ξω  = πω (x)ξω |T πω (a)† πω (y)ξω . Therefore, T πω (a)πω (x)ξω |πω (y)ξω  = T πω (x)ξω |πω (a)† πω (y)ξω ,

∀ a ∈ A, x, y ∈ A0 .

Hence, T ∈ (πω (A), Dω )w (see discussion before Proposition 2.2.6).

 

Lemma 3.2.6 Let ω, θ ∈ R(A, A0 ) with ω ≤ θ . Then, θ − ω ∈ R(A, A0 ). Proof The conditions (L.1) and (L.2) of Definition 2.4.6 are obviously satisfied. We check (L.3). For every a ∈ A and x ∈ A0 , using the Cauchy–Schwarz inequality for the positive sesquilinear form θ − ω , we obtain |(θ − ω)(a ∗ x)| = |(θ − ω )(a ∗ x, e)| = |(θ − ω )(x, a)| ≤ (θ − ω )(x, x)1/2 (θ − ω )(a, a)1/2 = (θ − ω )(a, a)1/2 (θ − ω)(x ∗ x)1/2 .

 

As mentioned at the beginning of this section, Rc (A, A0 ) denotes the set of all bounded representable linear functionals on (A[ · ], A0 ); i.e., ω ∈ Rc (A, A0 ), if and only if, there exists γ > 0, such that |ω(a)| ≤ γ a,

∀ a ∈ A.

It is easily seen that if ω ∈ Rc (A, A0 ), then for every x ∈ A0 , the linear functional ωx defined by ωx (a) := ω(x ∗ ax),

a ∈ A,

(3.2.19)

is bounded (see, for instance, (3.2.17)); thus ωx ∈ Rc (A, A0 ), for every x ∈ A0 .

3.2 Continuity of Representable Linear Functionals

59

Theorem 3.2.7 Let (A[ · ], A0 ) be a Banach quasi *-algebra with unit e. The following statements are equivalent: (i) every ω ∈ R(A, A0 ) is bounded; i.e., R(A, A0 ) = Rc (A, A0 ); (ii) every *-representation π of A[ · ] into L† (Dπ , Hπ )[tw ] is continuous; (iii) for every ω ∈ R(A, A0 ), ω = 0, there exists a nonzero θ ∈ Rc (A, A0 ), such that θ ≤ ω. Proof (i) ⇒ (ii) For the definition of tw , see end of Sect. 2.1.3. Let π be a *representation of (A[ · ], A0 ). Then, for every ξ ∈ Dπ the linear functional ω(a) := π(a)ξ |ξ , a ∈ A is representable, therefore bounded. This easily implies, using the polarization identity, that π is weakly continuous. (ii) ⇒ (iii) Let ω ∈ R(A, A0 ) and πω the corresponding GNS representation, which is weakly continuous by assumption and ξω the corresponding cyclic vector. Then, for every ξ, η ∈ Dω , there exists γξ,η > 0, such that |πω (a)ξ |η| ≤ γξ,η a,

∀ a ∈ A.

In particular, for the cyclic vector ξω , we have |ω(a)| = |πω (a)ξω |ξω | ≤ γξω ,ξω a,

∀ a ∈ A.

Then, (iii) holds with the obvious choice of θ = ω. (iii) ⇒ (i) By assumption, the set Kω = {θ ∈ Rc (A, A0 ) : θ ≤ ω} is a nonempty partially ordered (by ≤) set. Let W be a totally ordered subset of Kω . Then, limθ∈W θ (a) exists, for every a ∈ A. Indeed, the set of numbers {θ (a, a) : θ ∈ W} is increasing and bounded from above by ω (a, a). We set, ∀ a ∈ A.

(a, a) = lim θ (a, a), θ∈W

Then,  satisfies the equality (a + b, a + b) + (a − b, a − b) = 2(a, a) + 2(b, b),

∀ a, b ∈ A.

We can then define  on A × A using the polarization identity. Now we put ω◦ (a) := (a, e), a ∈ A. It is easy to see that ω◦ (a) = limθ∈W θ (a), a ∈ A. We prove that ω◦ ∈ Rc (A, A0 ). It is clear that ω◦ is a linear functional on A and ω◦ ≤ ω. The conditions (L.1) and (L.2) of Definition 2.4.6 are obviously satisfied. We prove (L.3). Let a ∈ A and x ∈ A0 . Then, |ω◦ (a ∗ x)| = lim |θ (a ∗ x)| ≤ lim (1 + θ (a, a))1/2 lim θ (x ∗ x)1/2 θ∈W

≤ (1 + (a, a))

θ∈W

1/2 ◦

θ∈W

∗

ω (x x)

1/2

.

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3 Normed Quasi *-Algebras: Basic Theory and Examples

We show now that ω◦ is bounded. For every a ∈ A the set {|θ (a)| : θ ∈ W} is bounded; indeed, for every θ ∈ W, we obtain |θ (a)| = |θ (a, e)| ≤ θ (a, a)1/2 θ (e, e)1/2 ≤ ω (a, a)1/2 ω (e, e)1/2 , a ∈ A. By the uniform boundedness principle, we conclude that there exists γ > 0, such that |θ (a)| ≤ γ a, for every θ ∈ W and for every a ∈ A. Hence, |ω◦ (a)| = lim |θ (a)| ≤ γ a, θ∈W

∀ a ∈ A.

Thus W has an upper bound. Then, by Zorn’s lemma, Kω has a maximal element. Let us call it ω• . It remains to prove that ω = ω• . Assume, on the contrary that ω > ω• . Let us consider the functional ω − ω• , which is representable by Lemma 3.2.6 and nonzero. Then, there exists σ ∈ Rc (A, A0 ), σ = 0, such that ω − ω• ≥ σ . Consequently, ω ≥ ω• + σ , contradicting the maximality of ω• . Thus, ω = ω• and   therefore ω is continuous. Remark 3.2.8 The equivalence of (i) and (ii) of the previous theorem holds also in the case when (A[ · ], A0 ) is only a normed quasi *-algebra. The proof of (iii) ⇒ (i) is similar to a known result in the theory of Banach *-algebras (see [7, Lemma 5.5.5]).

3.3 Continuity of *-Representations The seminorms p and q, introduced in Sect. 3.1.2 (see, (3.1.9) and (3.1.13), respectively), play an interesting role also in the study of the continuity of a *representation. As we shall see at the end of this section, the most favourable situation occurs when p is an m*-seminorm. In fact, in this case, A may be viewed (up to a quotient) as a subspace of the C*-algebra Ap and (as expected) any regular *-representation will be bounded and norm-continuous. But this is, in a sense, a rather extreme situation rarely realized in practice and Banach quasi *algebras having unbounded *-representations do really exist. For this reason, we begin with looking for conditions that guarantee the strong continuity of any regular *-representation. Proposition 3.3.1 Let A be a normed quasi *-algebra. The following statements hold: (i) every strongly continuous *-representation is regular; (ii) every completely regular *-representation is strongly continuous.

3.3 Continuity of *-Representations

61

Proof (i) Let π be strongly continuous. Then, for every ξ ∈ Dπ , there exists γξ > 0, such that π(a)ξ  ≤ γξ a,

∀ a ∈ A.

(3.3.20)

From the previous inequality and from the density of A0 in A it follows that, for every a ∈ A, π(a)ξ ∈ π(A0 )ξ . The statement then follows from Proposition 2.4.15. (ii) Let π be a completely regular *-representation of (A[ · ], A0 ) (see Definition 3.1.38). Then, for every ξ ∈ Dπ , the vector form ϕξ is bounded. Therefore, for some γξ > 0, π(a)ξ 2 = ϕξ (a, a) ≤ γξ a2 ,

∀ a ∈ A.  

Hence π is strongly continuous.

The statements (i) and (ii) of Proposition 3.3.1 are not, in general, equivalent unless s (A) = P (A), as the next theorem (see discussion after Proposition 2.4.16) IA A0 0 shows. Theorem 3.3.2 Let (A[ · ], A0 ) be a normed quasi *-algebra with unit e. The following statements are equivalent: s (A) is bounded; i.e., I s (A) = P (A); (i) every ϕ ∈ IA A0 A0 0 (ii) every regular *-representation π is completely regular; (iii) every regular *-representation π of (A[ · ], A0 ) is strongly continuous.

If (A[ · ], A0 ) is a Banach quasi *-algebra with unit, then (i), (ii) and (iii) are also equivalent to the following statement: s (A) is lower semicontinuous. (iv) every ϕ ∈ IA 0

Proof (i) ⇒ (ii) Let π be a regular *-representation of (A[·], A0 ). Then, for every s (A). Thus, by (i), ϕ is bounded. ξ ∈ Dπ , ϕξ ∈ IA ξ 0 (ii) ⇒ (iii) This follows immediately from (ii) of Proposition 3.3.1. s (A). Then π ◦ is a regular *-representation (see (iii) ⇒ (i) Let ϕ ∈ IA ϕ 0 Proposition 2.4.16). Hence, it is strongly continuous from our assumption. Thus, for some γϕ > 0, we have ϕ(a, a) = πϕ◦ (a)λϕ (e)|πϕ◦ (a)λϕ (e) = πϕ◦ (a)λϕ (e)2 ≤ γϕ a2 ,

∀ a ∈ A.

Finally, if A[ · ] is complete, then (iv) ⇒ (i) follows from Lemma 3.1.47. The implication (i) ⇒ (iv) is obvious.  

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3 Normed Quasi *-Algebras: Basic Theory and Examples

If π is strongly continuous, then by (3.3.20), we can define a new norm on Dπ by ξ π := sup π(a)ξ , a≤1

ξ ∈ Dπ .

Since (A[ · ], A0 ) has a unit, we have that ξ  ≤ ξ π , for every ξ ∈ Dπ . With this definition one has, of course, π(a)ξ  ≤ aξ π ,

∀ a ∈ A, ξ ∈ Dπ .

We put |||π(a)||| := sup π(a)ξ , ξ π ≤1

∀ a ∈ A.

By the definition itself it follows that |||π(a)||| ≤ a, for every a ∈ A. We denote with Repsc (A) the set of all strongly continuous *-representations of (A[ · ], A0 ). The next theorem shows that the seminorm p (see (3.1.9)) on A is determined by Repsc (A). Theorem 3.3.3 Let (A[ · ], A0 ) be a normed quasi *-algebra with unit e. Then, p(a) =

sup

π ∈ Repsc (A)

|||π(a)|||,

∀ a ∈ A.

Proof Let π ∈ Repsc (A). For any ξ ∈ Dπ we define, as before, ϕξ by ϕξ (a, b) = π(a)ξ |π(b)ξ ,

∀ a, b ∈ A.

Then, ϕξ ∈ PA0 (A), since from the discussion before present theorem, we conclude |ϕξ (a, b)| ≤ a b ξ 2π ,

∀ a, b ∈ A.

Clearly, ϕξ ∈ SA0 (A), if ξ π ≤ 1. Therefore, |||π(a)|||2 = sup π(a)ξ 2 = sup ϕξ (a, a) ≤ p(a)2 , ξ π ≤1

ξ π ≤1

∀ a ∈ A.

(3.3.21)

On the other hand, if ϕ ∈ SA0 (A) and πϕ is the corresponding GNS representation, then (see (2.4.8) and (2.3.5)) πϕ◦ (a)λϕ (x)2 = ϕ(ax, ax) = ϕx (a, a) ≤ ϕx  a2 ,

∀ a ∈ A and x ∈ A0 .

3.3 Continuity of *-Representations

63

Therefore, πϕ◦ is strongly continuous. Hence, λϕ (x)πϕ◦ = sup πϕ◦ (a)λϕ (x) = ϕx 1/2 , a≤1

∀ a ∈ A and x ∈ A0 ,

so   |||πϕ◦ (a)||| = sup πϕ◦ (a)λϕ (x) : x ∈ A0 with ϕx  ≤ 1 ,

a ∈ A.

This equality implies that sup

ϕ∈PA0 (A)

  |||πϕ◦ (a)||| = sup ϕx (a, a)1/2 : x ∈ A0 with ϕx  ≤ 1 =

sup

ϕ∈SA0 (A)

ϕ(a, a)1/2 = p(a),

∀ a ∈ A.

Consequently (see also (3.3.21)), p(a) =

sup

ϕ∈PA0 (A)

|||πϕ◦ (a)||| ≤

sup

π ∈ Repsc (A)

|||π(a)||| ≤ p(a),

∀ a ∈ A.

This completes the proof.

 

Example 3.3.4 A simple example of a Banach quasi *-algebra having a strongly continuous unbounded *-representation is provided by (Lp (I ), C(I )), I = [0, 1], with p ≥ 2. If we put (π(x)ξ )(t) = x(t)ξ(t), x ∈ Lp (I ), ξ ∈ C(I ), t ∈ I, then π is a *-representation of (Lp (I ), C(I )) in the Hilbert space L2 (I ). It is easily seen that π(x) is bounded, if and only if, x ∈ L∞ (I ). This *-representation is strongly continuous. Indeed, π(x)ξ 2 ≤ ξ ∞ x2 ≤ ξ ∞ xp , x ∈ Lp (I ), ξ ∈ C(I ). A criterion for (A[ · ], A0 ) to have only bounded strongly continuous *-representations is given by the following Proposition 3.3.5 Let (A[ · ], A0 ) be a Banach quasi *-algebra with unit e. The following statements are equivalent: (i) D(q) = A; (ii) every strongly continuous *-representation π of (A[ · ], A0 ) is bounded; (iii) every ϕ ∈ PA0 (A) is admissible, in the sense of Definition 2.4.4. Proof (i) ⇒ (ii) Assume that there exists a strongly continuous unbounded *representation π of (A[ · ], A0 ). Then, for some a ∈ A, π(a) is an unbounded

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3 Normed Quasi *-Algebras: Basic Theory and Examples

operator. This implies that there exists a sequence {ξn } of vectors in Dπ with the properties ξn  = 1,

π(a)ξn  → ∞. n→∞

As before, put ϕξn (b, c) = π(b)ξn |π(c)ξn , b, c ∈ A, n ∈ N. The strong continuity of π implies the existence of γ > 0, such that ϕξn (b, b) = π(b)ξn 2 ≤ γ b2 ξn 2 = γ b2 ,

∀ b ∈ A.

Thus, for every n ∈ N, ϕξn is bounded. As it is easily seen, ϕξn (e, e) = 1. Hence (see also (3.1.13)), q(a)2 ≥ sup ϕξn (a, a) = π(a)ξn 2 → ∞, a ∈ A. n∈N

n→∞

Consequently, a ∈ D(q), a contradiction. (ii) ⇒ (iii) Let ϕ ∈ PA0 (A). Then, for some γ > 0, ϕ(a, a) ≤ γ a2 , for every a ∈ A. If πϕ is the corresponding GNS representation, we obtain πϕ (a)λϕ (x)2 = ϕ(ax, ax) ≤ γ a2 x20 ,

∀ a ∈ A, x ∈ A0 .

Hence, πϕ is strongly continuous and thus bounded. The conclusion now results from Proposition 2.4.5. (iii) ⇒ (i) Suppose that there exists a ∈ A, such that a ∈ D(q). Then, πϕ (a)λϕ (e) = ϕ(a, a)1/2 ≥ q(a) ≥ ∞, which is a contradiction according to Proposition 2.4.5. This completes the proof.   Remark 3.3.6 If q(a) = 0 implies that a = 0, then by the previous proposition it follows that A is contained in the C*-algebra Aq , obtained by completing A0 with respect to the norm q. We don’t know if the identity map is necessarily continuous from A[ · ] into Aq [q]. If this is the case, then the next Proposition shows that p(a) = q(a), for every a ∈ A. Namely, we have the following Proposition 3.3.7 Let (A[ · ], A0 ) be a normed quasi *-algebra with unit e. The following statements are equivalent: (i) (ii) (iii) (iv) (v)

p is an m*-seminorm on A0 ; for each ϕ ∈ PA0 (A), ϕ = ϕ(e, e); D(q) = A and p(a) = q(a), for every a ∈ A; p is a C*-seminorm on A0 ; D(q) = A and q(a) ≤ a, for every a ∈ A.

3.3 Continuity of *-Representations

65

Proof (i) ⇒ (ii) Let ϕ ∈ PA0 (A). We define a linear functional  ωϕ on A0 /N0 (p) by  ωϕ (x + N0 (p)) = ϕ(x, e),

x ∈ A0 .

Then,  ωϕ is  · p -bounded and positive, since ϕ ∈ PA0 (A), respectively,

 ωϕ (x + N0 (p))∗ (x + N0 (p)) =  ωϕ (x ∗ x + N0 (p)) = ϕ(x ∗ x, e) = ϕ(x, x) ≥ 0,

x ∈ A0 .

We denote with  ωϕ the unique  · p -bounded extension of  ωϕ to Ap . If y ∈ Ap , then there exists a sequence {xn } ⊂ A0 , such that y =  · p − lim(xn + N0 (p)). Hence,

ωϕ (xn + N0 (p))∗ (xn + N0 (p)) ≥ 0.  ωϕ (y ∗ y) = lim  n→∞

ωϕ p of the linear Consequently,  ωϕ is positive on Ap . Then, for the norm  functional  ωϕ , one has ωϕ (e) = ϕ(e, e).  ωϕ p =  Therefore, for all x, y ∈ A0 , |ϕ(x, y)| = | ωϕ (y ∗ x + N0 (p))| ≤ ϕ(e, e)p(y ∗ x) ≤ ϕ(e, e)p(x)p(y) ≤ ϕ(e, e)xy. This inequality implies that ϕ ≤ ϕ(e, e) and since ϕ(e, e) ≤ ϕ, we obtain the equality. (ii) ⇒ (iii) follows immediately from the definition of q and from (ii). (iii) ⇒ (iv) follows from the equality p = q (see also (3.1.14)). (iii) ⇒ (v) immediately. (v) ⇒ (i) Since ϕ(ax, ax) ≤ q(a)2 ϕ(x, x),

∀ a ∈ D(q), x ∈ A0 ,

then ϕ(ax, ax) ≤ a2 ϕ(x, x),

∀ a ∈ A, x ∈ A0 .

For x = e this gives ϕ(a, a) ≤ a2 ϕ(e, e),

∀ a ∈ A.

Therefore, ϕ(e, e) ≥ ϕ. Since one always has ϕ(e, e) ≤ ϕ, we conclude that ϕ = ϕ(e, e). Thus, if ϕ(e, e) = 1, then ϕ ∈ SA0 (A). This implies that q(a) ≤

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3 Normed Quasi *-Algebras: Basic Theory and Examples

p(a), for every a ∈ A, by the very definitions. Hence, applying now Lemma 3.1.40,   (iii), we conclude that p is an m*-seminorm on A0 . Corollary 3.3.8 Let (A[ · ], A0 ) be a normed quasi *-algebra with unit e. The following statements are equivalent: (i) p is an m*-seminorm on A0 ; (ii) every regular *-representation π of (A[ · ], A0 ) in a Hilbert space H is bounded and continuous from A[ · ] into B(H), such that π(a) ≤ a, for every a ∈ A. Proof It follows immediately from Propositions 3.1.39 and 3.3.7.

 

As seen in previous examples, there exist Banach quasi *-algebras (A[ · ], A0 ) (with unit) for which PA0 (A) = {0}. If this is the case, there is no strongly continuous *-representation of (A[ · ], A0 ), apart from the trivial one. This unpleasant feature is prevented if we require that PA0 (A) is sufficient by which we mean that if a ∈ A and ϕ(a, a) = 0, for every ϕ ∈ PA0 (A), then a = 0. Clearly, if PA0 (A) is sufficient, then (A[ · ], A0 ) has a sufficient family of strongly continuous *-representations, where sufficient means in this case, that for every a ∈ A, a = 0, there exists a strongly continuous *-representation π , such that π(a) = 0. In particular, if p(a) = a, for every a ∈ A, then PA0 (A) is clearly sufficient and, as shown in [80, Theorem 3.26], D(q) is a C*-algebra, consisting of all bounded elements of A (see Example 3.1.45). In this case, it follows from Proposition 3.3.5, that a genuine Banach quasi *algebra (A[ · ], A0 ), in the sense, that A  A0 , necessarily has strongly continuous unbounded *-representations. Banach quasi *-algebras, with a sufficient PA0 (A), have been studied in more details in [49] and in [80]. The question, of characterizing in terms of the original norm of A, the existence of sufficiently many positive invariant sesquilinear forms, still remains open.

Chapter 4

Normed Quasi *-Algebras: Bounded Elements and Spectrum

Bounded elements of a Banach quasi *-algebra are intended to be those, whose images under every *-representation are bounded operators in a Hilbert space. This rough idea can be developed in several ways, as we shall see in the present chapter. These notions lead us to discuss a convenient concept of spectrum of an element in this context.

4.1 The *-Algebra of Bounded Elements 4.1.1 Bounded Elements Definition 4.1.1 Let (A[ · ], A0 ) be a Banach quasi *-algebra and a ∈ A. We say that a is left bounded if there exists γa > 0, such that ax ≤ γa x,

∀ x ∈ A0 .

The set of all left bounded elements of A is denoted by Alb . Analogously, we say that a is right bounded if there exists γa > 0, such that xa ≤ γa x,

∀ x ∈ A0 .

The set of all right bounded elements of A is denoted by Arb . The motivation of this terminology is made clearer by considering, for each a ∈ A, the linear maps (3.1.12) from A0 into A: x ∈ A0 → La x := ax ∈ A, x ∈ A0 → Ra x := xa ∈ A.

© Springer Nature Switzerland AG 2020 M. Fragoulopoulou, C. Trapani, Locally Convex Quasi *-Algebras and their Representations, Lecture Notes in Mathematics 2257, https://doi.org/10.1007/978-3-030-37705-2_4

(4.1.1)

67

68

4 Normed Quasi *-Algebras: Bounded Elements and Spectrum

For instance, if a is left bounded, then La is bounded, as a map from A0 [ · ] into A[ · ], and so it has a bounded extension La to A[ · ]. We put alb := La , a ∈ A. Analogously, we define a norm on Arb by arb := R a , a ∈ A. Remark 4.1.2 Observe that an element a ∈ Alb is not necessarily right bounded. We put Ab = Alb ∩ Arb . On Ab we define the norm   ab := max La , R a  , a ∈ Ab .

(4.1.2)

It is easily seen that, if a ∈ Ab , then a ∗ ∈ Ab . Moreover, by the very definitions, (3.1.4) and the discussion around it, we see that A0 ⊆ Ab , xb = x0 , for all x ∈ A0 and also an analogous result to that of Corollary 3.1.3 holds for  · b . Remark 4.1.3 If (A[ · ], A0 ) has a unit e, then obviously e ∈ Ab and since we may suppose that e = 1 (Remark 3.1.4), we obtain alb = La  ≥ a

∀ a ∈ Alb .

Of course, an analogous inequality holds for  · rb . Thus, it turns out that, in the unital case,  ·  ≤  · b , on Ab . As usual, we denote with B(A) the Banach algebra of bounded operators on the Banach space A[ · ]. From the definition itself it follows that Alb , as well as Arb , can be identified with a subspace of B(A). Let a ∈ Alb and b ∈ A. Then, we put a b := La b.

(4.1.3)

More explicitly, a b is defined by picking a sequence {yn } ⊂ A0 converging to b and setting a b := La b = lim La yn = lim ayn . n→∞

n→∞

Similarly, if b ∈ Arb and a ∈ A, we put a b := R b a.

(4.1.4)

4.1 The *-Algebra of Bounded Elements

69

In this case, for an explicit definition of a b, we have to select a sequence {xn } ⊂ A0 converging to a and put a b := R b a = lim Rb xn = lim xn b. n→∞

n→∞

Then, it is clear that, if a, b ∈ Ab , we have that both a b and a b are well defined, but, in general, a b = a b. We shall discuss this point, in more details, below. Remark 4.1.4 Let a, b ∈ Alb . Then, La , Lb ∈ B(A). Hence, La Lb ∈ B(A) and La Lb = La b . Indeed, let {yn } be a sequence in A0 ,  · -converging to b. Then, for each x ∈ A0 , (La Lb )x = La (Lb x) = lim a(yn x) = La (bx). n→∞

On the other hand, La b x = (La b)x = lim (ayn )x = lim a(yn x) = La (bx), x ∈ A0 . n→∞

n→∞

(4.1.5)

It is easy to show that if a, b ∈ Alb and μ ∈ C, then both a + b and μa belong to Alb . Proposition 4.1.5 If (A[ · ], A0 ) is a Banach quasi *-algebra with unit, then the set Alb of all left bounded elements is a Banach algebra, with respect to the multiplication  and the norm  · lb . Proof We prove that if a, b ∈ Alb then a b ∈ Alb and a blb ≤ alb blb . Indeed, using the associativity properties of the multiplication in A, one has that (La b)x = lim (aym )x = lim a(ym x) = La (bx) = La (Lb x), m→∞

m→∞

∀ x ∈ A0 ,

where {ym } is a sequence in A0 ,  · -converging to b. Therefore, we have (see Remark 4.1.4) La b x = (La b)x ≤ La  Lb  x,

∀ x ∈ A0 .

Hence, a b ∈ Alb and moreover (see also discussion after (4.1.1)), we obtain La b  ≤ La  Lb  ≤ alb b lb . From the latter inequality, together with (4.1.3) and (4.1.5), it follows that a blb = La b  ≤ alb blb ,

∀ a, b ∈ Alb .

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4 Normed Quasi *-Algebras: Bounded Elements and Spectrum

Thus, Alb , endowed with  · lb , is a normed algebra. We shall now show that Alb [ · lb ] is complete. Let {an } be a Cauchy sequence in Alb [ · lb ]. Then, {Lan } is a Cauchy sequence in B(A). Thus, there exists L ∈ B(A), such that Lan → L, with respect to the natural norm of B(A). By Remark 4.1.3, an − am  → 0, so there exists a ∈ A, such that an − a → 0. Since, the right multiplication by x is n→∞

continuous in A, it follows that an x → ax = La x, in the norm of A. This implies n→∞

that La = L. From these facts, it follows easily that a is left bounded and an → a, n→∞ with respect to  · lb .   A similar result can be proved for Arb , taking into account the following facts concerning the involution * of A: (1*) a ∈ Alb ⇔ a ∗ ∈ Arb ; (2*) a ∗ rb = alb , for every a ∈ Alb ; (3*) (a b)∗ = b∗ a ∗ , for every a, b ∈ Alb . These facts, as well as the next lemma, follow easily by the very definitions. Lemma 4.1.6 If (A[·], A0 ) is a Banach quasi *-algebra, the following statements hold: (i) if a ∈ Alb and b ∈ A, then a b ≤ alb b; (ii) if b ∈ Arb and a ∈ A, then a b ≤ abrb .

4.1.2 Normal Banach Quasi *-Algebras As mentioned before, if a, b ∈ Ab then, both a b and a b are well defined; but, in general, a b = a b. We want to analyze, more carefully, this situation. First of all, if a, b ∈ Ab , then La , Lb ∈ B(A). As shown in Remark 4.1.4, La Lb = La b . Similarly, if a, b ∈ Ab , then R b R a = R a b . Recall that we denote with A the Banach dual space of A[ · ] and by f  the norm of f ∈ A . Proposition 4.1.7 If (A[ · ], A0 ) is a Banach quasi *-algebra, the following statements are equivalent: (i) (ii) (iii) (iv)

a b = a b, for every a, b ∈ Ab ; a b is right bounded and a b ≤ abrb , for every a, b ∈ Ab ; a b is left bounded and a b ≤ alb b, for every a, b ∈ Ab ; for any pair {xn }, {yn } of sequences of elements of A0 ,  · -converging to elements of Ab , one has lim lim xn ym = lim lim xn ym ;

n→∞ m→∞

m→∞ n→∞

4.1 The *-Algebra of Bounded Elements

71

(v) there exists a weakly*-dense subspace M of A , such that for any pair {xn }, {yn } of sequences of elements of A0 ,  · -converging to elements of Ab , one has lim lim f (xn ym ) = lim lim f (xn ym ),

n→∞ m→∞

m→∞ n→∞

∀ f ∈ M.

Proof (i) ⇒ (ii) It follows directly from the assumption a b = a b in (i) and Lemma 4.1.6(ii). (ii) ⇒ (iii) Use the fact that a ∗  = a, for all a ∈ A and apply (ii) and (1*), (2*), (3*), before Lemma 4.1.6. (iii) ⇒ (i) Assume that, for every a, b ∈ Ab , a b is left bounded and a b ≤ alb b. Let {yn } ⊂ A0 be a sequence, such that b − yn  → 0, as n → ∞. Then, since A0 ⊆ Ab and Ab is a vector space, we get a b − ayn  = a b − a yn  = a (b − yn ) ≤ alb b − yn  → 0. Hence, a b = lim ayn = La b = a b. n→∞

(i) ⇒ (iv) Let {xn }, {yn } be sequences of elements of A0 , with a − xn  → 0, b − yn  → 0, as n → ∞, where a, b ∈ Ab . Then, we have a b = La b = lim aym = lim lim xn ym . m→∞

m→∞ n→∞

On the other hand, a b = R b a = lim xn b = lim lim xn ym . n→∞

n→∞ m→∞

The equality a b = a b then implies that the two iterated limits coincide. (iv) ⇒ (v) It is clear. (v) ⇒ (i) Assume that (i) fails. Then, there exists f ∈ A , such that f (a b) = f (a b). Since M is weakly*-dense in A , we may suppose that f ∈ M. Then, if {xn }, {yn } are sequences in A0 ,  · -converging, respectively, to a and b, we have lim lim f (xn ym ) = f (a b) = f (a b) = lim lim f (xn ym ),

m→∞ n→∞

n→∞ m→∞

 

a contradiction. This completes the proof. If anyone of the equivalent conditions of Proposition 4.1.7 holds, we put a • b := a b = a b, and a • b is called the product of a and b.

a, b ∈ Ab ,

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4 Normed Quasi *-Algebras: Bounded Elements and Spectrum

Definition 4.1.8 A Banach quasi *-algebra (A[ · ], A0 ), such that a b = a b, for all a, b in Ab , is called normal. Corollary 4.1.9 If (A[ · ], A0 ) is a Banach quasi *-algebra with unit, the following statements hold: (i) (A[ · ], A0 ) is normal, if and only if, Ab is a *-algebra, with respect to  (or, equivalently, with respect to ); (ii) if (A[ · ], A0 ) is a normal Banach quasi *-algebra, then Ab [ · b ] is a Banach *-algebra, with respect to the multiplication •. Proof (i) The fact that if (A[·], A0 ) is normal, then Ab is a *-algebra, with respect to , follows from the previous discussion. On the other hand, assume that Ab is a *-algebra, with respect to ; then, for every a, b ∈ Ab , a b ∈ Ab and (see also (3*), before Lemma 4.1.6) a b = (b∗ a ∗ )∗ = a b. (ii) It follows easily from Proposition 4.1.5 and the properties of involution.

 

Example 4.1.10 Assume that for each a ∈ Ab there exists a sequence {xn } ⊂ A0 , such that sup xn 0 < ∞ and n

lim a − xn  = 0;

n→∞

then (A[ · ], A0 ) is normal. Indeed, in this case, (ii) or (iii) of Proposition 4.1.7 holds (see Proposition 4.1.19). Elements of the preceding type will be called strongly bounded and will be studied in Sect. 4.1.3. Remark 4.1.11 If (A[·], A0 ) is a commutative Banach quasi *-algebra, i.e., ax = xa, for every a ∈ A and x ∈ A0 , then it is easily seen that each left bounded element a is also right bounded and one has a b = ba, for every a, b ∈ Ab . Thus, if a, b ∈ Ab , then both a b and a b are elements of Ab , but they need not be equal. In general, in such a case, Ab is an algebra with respect to  (and also with respect to ). Normality, in the commutative case, becomes equivalent to commutativity of the algebra Ab . Example 4.1.12 For the Banach quasi *-algebra (Lp (I ), C(I )) considered in Example 3.1.6, one finds that (Lp (I ))b = L∞ (I ) and the norm  · b is exactly the L∞ -norm. Since the multiplications  and  both coincide with the ordinary multiplication of functions, (Lp (I ), C(I )) is normal. This example also shows that, in general, A0 is not dense in Ab , with respect to  · b since, as is well known, C(I ) is not dense in L∞ (I ). Similarly, (Lp (R), Cc0 (R)) is a Banach quasi *-algebra without unit. In this case, p (L (R))b = L∞ (R) ∩ Lp (R) and (Lp (R), Cc0 (R)) is normal. The norm  · b is equivalent to  · p +  · ∞ .

4.1 The *-Algebra of Bounded Elements

73

For the non-commutative Lp -spaces of Example 3.1.7 one finds that (Lp ())b = Jp , if  is semifinite, or (Lp ())b = M, if  is finite. Normality follows from the fact that the multiplications  and  both coincide with the ordinary multiplication of bounded operators. Example 4.1.13 In case of the Banach quasi *-algebra (H, A0 ) constructed from a Hilbert algebra A0 , as in Example 3.1.8, the set Hb of bounded elements of H is the so-called fulfillment of A0 (A0 is called a full Hilbert algebra if Hb = A0 ). Moreover, (H, A0 ) is normal. Indeed, let a, b ∈ Hb , and let {xn }, {yn } be sequences in A0 ,  · -converging, respectively, to a and b. Then, a b|x = lim ayn |x = lim yn |a ∗ x = b|a ∗ x, n→∞

n→∞

∀ x ∈ A0 .

On the other hand, a b|x = lim xn b|x = lim b|xn∗ x = b|a ∗ x, n→∞

n→∞

∀ x ∈ A0 .

This implies that a b = a b. Note that, in general, a Banach quasi *-algebra (A[ · ], A0 ) is said to be full if Ab = A0 . Example 4.1.14 Let A be an unbounded densely defined selfadjoint operator in L2 (), where  is an open subset of Rn . Let D(A) denote the domain of A. Then, D(A) can be made into a Hilbert space under the inner product f |gA = f |g + Af |Ag,

f, g ∈ D(A)

1/2

and norm  · A = ·|·A . Let us assume that (i) Cc () ⊂ D(A) and that it is a core for A, i.e., A Cc () = A [2, Definition 1.4.1], (here Cc () stands for the algebra of continuous functions on , with compact support); (ii) f ∈ D(A) implies f ∗ ∈ D(A), where f ∗ (x) = f (x); (iii) (Af )∗ = Af ∗ , for every f ∈ D(A); (iv) f φ ∈ D(A), for every f ∈ D(A) and φ ∈ Cc (). Then (D(A), Cc ()) is a Banach quasi *-algebra, when D(A) is endowed with  · A . The continuity of the multiplication follows from the closed graph theorem and the fact that, for each φ ∈ Cc (), the multiplication operator Lφ is closed in the Hilbert space D(A). The space of bounded elements is given by the set   DM (A) = f ∈ D(A) : f g ∈ D(A), ∀ g ∈ D(A) .

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4 Normed Quasi *-Algebras: Bounded Elements and Spectrum

Indeed, it is clear that for any bounded element f , the operator Lf maps D(A) into D(A). Conversely, if f ∈ DM (A), then the corresponding operator Lf is closed in the Hilbert space D(A) and therefore bounded. Hence, D(A)b = DM (A). To conclude this example we notice that if Lφ A = ALφ , for every φ ∈ Cc (), then Cc () is a Hilbert algebra with inner product ·|·A . By Example 4.1.13 it follows that (D(A), Cc ()) is normal. Lemma 4.1.15 If (A[ · ], A0 ) is a normal Banach quasi *-algebra, then, La R b = R b La ,

∀ a, b ∈ Ab .

(4.1.6)

Proof Indeed, let a, b ∈ Ab , and {xn }, {yn } be sequences in A0 ,  · -converging, respectively, to a and b. Then, for every x ∈ A0 , (La R b )x = La (R b x) = lim a(xym ) = lim lim xn (xym ). m→∞

m→∞ n→∞

On the other hand, (R b La )x = R b (La x) = lim (xn x)b = lim lim (xn x)ym . n→∞

n→∞ m→∞

The statement then follows from (iv) of Proposition 4.1.7.

 

Remark 4.1.16 If (A[ · ], A0 ) has a unit, then (4.1.6) implies the normality of (A[ · ], A0 ). If (A[·], A0 ) is a normal Banach quasi *-algebra the multiplications of an element of A and an element of Ab are defined via (4.1.3) and (4.1.4). More precisely,  a•b =

a b, if a ∈ Ab a b, if b ∈ Ab .

Proposition 4.1.17 If (A[ · ], A0 ) is a normal Banach quasi *-algebra, then (A, Ab ) is a BQ*-algebra. Proof Let a, b ∈ A, with a ∈ A, b ∈ Ab . Then, if {xn } is a sequence in A0 converging to a, we get (a • b)∗ = (a b)∗ = (Rb a)∗ = lim (xn b)∗ = lim b∗ xn∗ = Lb∗ a ∗ = b∗ a ∗ = b∗ • a ∗ . n→∞

n→∞

The case a ∈ Ab , b ∈ A is similar. For a ∈ A, b ∈ Ab , Lemma 4.1.6 implies that a • b ≤ abb .

4.1 The *-Algebra of Bounded Elements

75

It remains only to check the module associativity rules. Let a ∈ A and b1 , b2 ∈ Ab . Then, a • (b1 • b2 ) = a (b1 • b2 ) = a (b1 b2 ) = R b1 b2 a = (R b2 R b1 )a = R b2 (R b1 a) = (a b1 )b2 = (a • b1 ) • b2 . Using (4.1.6), we also have (b1 • a) • b2 = (b1 a)b2 = R b2 (b1 a) = R b2 (Lb1 a) = (R b2 Lb1 )a = (Lb1 R b2 )a = Lb1 (R b2 a) = Lb1 (a b2 ) = b1 (a b2 ) = b1 • (a • b2 ). The fact that, Ab [ · b ] is a Banach algebra (see Corollary 4.1.9(ii)) completes the proof.  

4.1.3 Strongly Bounded Elements Throughout this subsection (A[ · ], A0 ) will be a Banach quasi *-algebra. In Example 4.1.10 we anticipated the definition of a strongly bounded element, which we want to discuss here in more details [81, 84]. We repeat, for convenience the definition. Definition 4.1.18 An element a ∈ A is called strongly bounded if there exists a sequence {xn } ⊂ A0 , such that: sup xn 0 < ∞ and n

lim a − xn  = 0.

n→∞

The set of strongly bounded elements is denoted with Asb . Proposition 4.1.19 Each strongly bounded element in A is bounded. If a, b ∈ Asb , then a b = a b. Proof Let a be strongly bounded and {xn } ⊂ A0 a sequence satisfying the conditions of Definition 4.1.18. Then, for every x ∈ A0 , we have xa = lim xxn  ≤ sup xn 0 x, resp. ax ≤ sup xn 0 x. n→∞

n

n

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4 Normed Quasi *-Algebras: Bounded Elements and Spectrum

Thus, since a is strongly bounded, Definition 4.1.1 is fulfilled, therefore a is bounded. Let a, b ∈ Asb and {xn }, {yn }  · 0 -bounded sequences of elements from A0 ,  · -converging, to the elements a and b of A, respectively. Then, we have a b = La b = lim La ym = lim lim xn ym . m→∞

m→∞ n→∞

Similarly, a b = R b a = lim Rb xm = lim lim xm yn . m→∞

m→∞ n→∞

Then, a b − a b = lim lim xn ym − xm yn . m→∞ n→∞

But xn ym − xm yn  ≤ xn ym − xm ym  + xm ym − xm yn  ≤ sup ym 0 xn − xm  + sup xm 0 ym − yn  → 0. m

m

 

Therefore, a b = a b.

Example 4.1.20 An example of a Banach quasi *-algebra, where every bounded element is strongly bounded is provided by Hilbert algebras (H, A0 ) (see Example 3.1.8). For this result we refer to [19, Proposition 11.7.9]. Of course, this statement applies also to the situation described at the end of Example 4.1.14. Proposition 4.1.21 The space Asb , endowed with the norm  · b , is a Banach *algebra. Proof It is easily seen that Asb is stable under addition and multiplication by scalars. Let a, b ∈ Asb . If {xn } and {yn } are  · 0 -bounded sequences converging, respectively, to a and b in A, we have a • b − xn yn  ≤ a • b − ayn  + ayn − xm yn  + xm yn − xn yn  ≤ a • b − ayn  + sup yn 0 a − xm  + sup yn 0 xm − xn , n

n

which goes to 0, for n, m → ∞. It is clear that the sequence {xn yn } is ·0 -bounded. To prove the completeness, it is sufficient to show that Asb is closed in Ab . Let a ∈ Asb . Then, for every ε > 0 there exists aε ∈ Asb such that a − aε  < ε/2. On the other hand, by Definition 4.1.18, we can find a sequence {xn } ⊂ A0 , such that sup xn 0 < ∞ and n

lim aε − xn  = 0.

n→∞

4.1 The *-Algebra of Bounded Elements

77

Hence, there exists nε ∈ N, in such a way that, for every n ≥ nε , aε − xn  < ε/2. Thus, in conclusion, for n ≥ nε , a − xn  ≤ a − aε  + aε − xn  < ε, consequently, a ∈ Asb .

 

Of course, if Asb = Ab , then by Proposition 4.1.19, (A[ · ], A0 ) is normal (see discussion before Proposition 4.1.5). In order to provide some sufficient condition for this equality to hold, we study some commutation properties of (A[ · ], A0 ). If M ⊆ B(A), we denote with Mc the commutant of M, that is   Mc = X ∈ B(A) : XY = Y X, ∀ Y ∈ M . Let us consider the sets c  L = Rx : x ∈ A0 ,

c  R = Lx : x ∈ A0 .

It is easily seen that both L and R are subalgebras of B(A). For every x, y ∈ A0 one has Lx Ry = Ry Lx . Hence, {Lx : x ∈ A0 } ⊂ L and thus Lc ⊆ R. We shall prove that the last inclusion is, in fact, equality (see Proposition 4.1.26). Lemma 4.1.22 The following statements hold: (i) if a is left bounded and S ∈ L, then Sa is left bounded and LSa = SLa ; (ii) if a is right bounded and T ∈ R, then T a is right bounded and R T a = T R a . Proof We prove only (i); (ii) is similarly proved. For every x ∈ A0 one has LSa x = (Sa)x = Rx Sa = (Rx S)a = SRx a = Sax. This implies that (Sa)x = Sax ≤ S ax ≤ S γa x ≤ γ x, for all x ∈ A0 and γ = Sγa (see Definition 4.1.1).

 

Lemma 4.1.23 If a is left, respectively, right bounded, then La ∈ L, respectively, R a ∈ R. Proof Using the module associativity one has, for every x ∈ A0 , La Rx y = La yx = a(yx) = Rx La y, Since A0 is dense in A, the equality extends to A.

∀ y ∈ A0 .  

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4 Normed Quasi *-Algebras: Bounded Elements and Spectrum

Lemma 4.1.24 Let T ∈ R. Then, T La = La T , for every left bounded element a ∈ A. Proof Let a be left bounded and {xn } a sequence in A0 converging to a. Then, for every y ∈ A0 , we have T La y = T ay = T ( lim xn y) n→∞

= lim T xn y = lim T Lxn y = lim Lxn T y. n→∞

n→∞

n→∞

On the other hand, since each y ∈ A0 is right bounded (see discussion at the beginning of Sect. 3.3) and T ∈ R, we can use (ii) of Lemma 4.1.22 and obtain La T y = R T y a = lim RT y xn = lim Lxn T y. n→∞

n→∞

  In particular, we have Corollary 4.1.25 If a is left bounded and b is right bounded, then La R b = R b La . Proof From Lemma 4.1.23, one has that R b ∈ R; so the statement follows from   Lemma 4.1.24. Proposition 4.1.26 If either (A[ · ], A0 ) has a unit e or A20 is total in A, we have Lc = R,

Rc = L.

Proof It is sufficient to prove that each S ∈ L commutes with each T ∈ R. This implies, in fact, that L ⊆ Rc . But, as noticed before, the converse inclusion always holds and so we obtain the equality. Let x, y ∈ A0 . Taking into account that each x ∈ A0 is left bounded, Sx is left bounded (Lemma 4.1.22) and LSx = SLx ; yet Lemma 4.1.24 can be applied to obtain ST xy = ST Lx y = SLx T y = LSx T y = T LSx y = T Sxy. Since either A20 = A0 or A20 is total in A, the claim follows. The previous Proposition implies that

Lcc

= L and

Rcc

= R.

Proposition 4.1.27 Assume that (A[ · ], A0 ) has a unit e. Then,   L = La : a ∈ Alb ,

  R = R a : a ∈ Arb .

 

4.1 The *-Algebra of Bounded Elements

79

Proof We know that La ∈ L, for every a ∈ Alb . Now assume that X ∈ L and put a = Xe. We prove that X = LXe . Indeed, for every x, y ∈ A0 , we obtain Xxy = XRy x = (XRy )x = (Ry X)x = (Xx)y. For x = e we have Xy = (Xe)y, for every y ∈ A0 . Hence, (Xe)y = Xy ≤ X y,

∀ y ∈ A0 .

That is, Xe is left bounded. The equality concerning R is similarly proved.

 

Before going forth we need some further notation and some definitions. In what follows, if E[ · E ] is a normed space we denote with U (E) its unit ball, i.e., U (E) = {a ∈ E : aE ≤ 1}. The strong topology in the algebra B(E) of bounded operators on E is defined by the seminorms S ∈ B(E) → SaE ∈ R+ , a ∈ E. Proposition 4.1.28 Let (A[ · ], A0 ) have a unit e. Consider the following statements:  



(i) Lx : x ∈ A0  ∩ U B(A) is strongly dense in L ∩ U B(A) ; (ii) Rx : x ∈ A0 ∩ U B(A) is strongly dense in R ∩ U B(A) ; (iii) Asb = Alb = Arb = Ab . Then, (i) ⇔ (ii) ⇒ (iii)



Proof (i) ⇒ (ii) By Proposition 4.1.27, if S ∈ R ∩ U B(A) , then S = R a , for

some a ∈ Arb , with arb ≤ 1. Since a ∗ ∈ Alb , then La ∗ ∈ L ∩ U B(A) . Therefore, there exists a sequence {xn } ⊂ A0 , with xn 0 ≤ 1, such that La ∗ b − Lxn b = a ∗ b − xn b → 0, n→∞

∀ b ∈ A.

The continuity of the involution in A implies that b∗ a − b∗ xn∗  = R a b∗ − Rxn∗ b∗  → 0,

∀ b ∈ A.

(ii) ⇒ (i) can be proven in the very same way.

(i) ⇒ (iii) Let a ∈ Alb . Put b = a/alb . Then, Lb ∈ L ∩ U B(A) . Therefore, there exists a sequence {xn } ⊂ A0 , with xn 0 ≤ 1, such that Lb c − Lxn c → 0,

∀ c ∈ A.

For c = e, b − xn  → 0. Thus, the sequence {alb xn } converges to a and xn 0 ≤ alb , for every n ∈ N. Therefore, a ∈ Asb .  

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4 Normed Quasi *-Algebras: Bounded Elements and Spectrum

Remark 4.1.29 We note that if (A[ · ], A0 ) has no unit, it can be proved, with a proof similar to that of Proposition 11.7.9 in [19], that (i) and (ii) imply (iii), but (i) and (ii) are no more equivalent, in general. To conclude, we observe that (i) or (ii) closely reminds of Kaplansky’s density theorem and this is the reason why, for the Banach quasi *-algebra (H, A0 ) constructed from a Hilbert algebra A0 , as in Example 3.1.8, the set of bounded and strongly bounded elements coincide. In fact, in that case, L and R are both von Neumann algebras and Kaplansky’s density theorem can be applied.

4.2 The Spectrum of an Element In this section we will discuss how the notion of spectrum of an element, which is fundamental in the theory of Banach algebras, can be extended to Banach quasi *-algebras.

4.2.1 The Inverse of an Element We first come back to a closer analysis of the linear maps La , Ra defined in (3.1.12). In general they are unbounded maps in the Banach space A. It is natural to deal with the problem of inverting an element a ∈ A first by inverting them. As customary in the theory of unbounded operators, we will look for bounded inverses. Definition 4.2.1 Let (A[ · ], A0 ) be a Banach quasi *-algebra with unit e. An element a ∈ A is called closable if the linear maps La : x ∈ A0 → ax ∈ A,

Ra : x ∈ A0 → xa ∈ A

are closable in A. Remark 4.2.2 A Banach quasi *-algebra, such that every a ∈ A is closable in A, was called fully closable in [49]. As we have done for bounded elements (see discussion before and after Remark 4.1.3), if a ∈ A we denote by La the closure of La , i.e., the linear operator defined on,   D(La ) := b ∈ A : ∃ {xn } ⊂ A0 : b − xn  → 0 and {axn } is Cauchy . by La b = lim axn . n→∞

4.2 The Spectrum of an Element

81

Similarly, R a will denote the closure of Ra and D(R a ) its domain. The definitions are obvious modifications of the previous ones. Lemma 4.2.3 Let a be a closable element of (A[·], A0 ). The following statements hold: (i) if y ∈ D(La ) and x ∈ A0 , then yx ∈ D(La ) and La yx = (La y)x; (ii) if y ∈ D(R a ) and x ∈ A0 , then xy ∈ D(R a ) and R a xy = x(R a y); (iii) D(R a ) = D(La ∗ )∗ and La ∗ y ∗ = (R a y)∗ , for every y ∈ D(R a ); (iv) if La has an everywhere defined inverse, then

(La )−1 c x = (La )−1 cx,

∀c ∈ A, x ∈ A0 ;

(v) if R a has an everywhere defined inverse, then

x (R a )−1 c = (R a )−1 xc,

∀ c ∈ A, x ∈ A0 .

Proof We begin with proving (i). Let y ∈ D(La ). Then, there exists a sequence {yn } ⊂ A0 such that y − yn  → 0 and {ayn } is Cauchy. Clearly, if x ∈ A0 , yn x − yx → 0 and, since a(yn x) − a(ym x) = a(yn − ym )x ≤ a(yn − ym )x0 , the sequence {a(yn x)} is Cauchy too. Thus, yx ∈ D(La ) and La (yx) = lim a(yn x) = lim (ayn )x = (La y)x. n→∞

n→∞

The proof of (ii) is similar and we omit it. As for (iii), let y ∈ D(R a ). Then there exists a sequence {yn } ⊂ A0 such that y = limn→∞ yn and limn→∞ yn a = z =: R a y. This implies that y ∗ = limn→∞ yn∗ and limn→∞ a ∗ yn∗ = z∗ =: La ∗ y ∗ . Hence y ∗ ∈ D(La ∗ ) and La ∗ y ∗ = (R a y)∗ . The inclusion D(La ∗ )∗ ⊂ D(R a ) can be proved in analogous way. −1 Now we prove (iv). If c ∈ A, there exists ∈ D(L z −1

a ) such that (La ) c = z. Then by (i), we get that, for every x ∈ A0 , (La ) c x = zx ∈ D(La ). On the other hand, since zx ∈ D(La ) there exists a unique w ∈ A, such that zx = ((La )−1 c)x = (La )−1 w. Now, by applying La to both sides of this equality we obtain

La (zx) = La (La )−1 c x = La (La )−1 cx = cx.

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4 Normed Quasi *-Algebras: Bounded Elements and Spectrum

Hence,

(La )−1 c x = (La )−1 cx,

∀ c ∈ A, x ∈ A0 .  

The proof of (v) is similar and we omit it.

Definition 4.2.4 Let (A[ · ], A0 ) be a Banach quasi *-algebra with unit e and a ∈ A a closable element. We say that a has a bounded inverse if there exists b ∈ Ab ∩ D(La ) ∩ D(R a ), such that R b a = Lb a = e. If (A[ · ], A0 ) is normal, then this element b, if any, is unique by (4.1.6). In this case, we denote the bounded inverse of a by a −1 . For a Banach (*-)algebra, the existence of the inverse of an element a can be characterized through the invertibility of the corresponding maps La , Ra . A similar result holds also in our case. Proposition 4.2.5 Let (A[ · ], A0 ) be a normal Banach quasi *-algebra with unit e and a ∈ A a closable element. The following statements are equivalent: (i) the element a has a bounded inverse; (ii) both La and R a possess everywhere defined (hence, bounded) inverses. Proof (i) ⇒ (ii) Suppose that there exists b ∈ Ab ∩ D(La ) ∩ D(R a ), such that R b a = Lb a = e. We prove that La is injective and surjective. Let y ∈ D(La ) with La y = 0. By definition there exists a sequence {yn } ⊂ A0 such that yn → y and ayn → La y = 0. Since b is bounded, we have Lb ayn → 0. By (i) of Lemma 4.2.3, we get y = lim yn = lim (Lb a)yn = lim Lb ayn = 0. n→∞

n→∞

n→∞

Therefore La is injective. Let now c ∈ A. Then, c is the limit of a sequence {zn } ⊂ A0 . Let yn := bzn and y = limn→∞ yn . Again by (i) of Lemma 4.2.3 it follows that bzn ∈ D(La ), for every n ∈ N. Taking into account the assumption of normality we have La bzn = (La b)zn = (R b a)zn = zn → c. Hence, y ∈ D(La ) and La y = c. This proves that La is surjective. Therefore (La )−1 exists, it is everywhere defined and closed, hence bounded. With a similar argument, taking into account that b ∈ D(R a ), one proves that R a has a bounded inverse. (ii) ⇒ (i) Let us now assume that La and R a have bounded inverses (La )−1 and (R a )−1 , respectively. Let us define b = (La )−1 e and b = (R a )−1 e. Then b ∈ D(La ) and b ∈ D(R a ) and, clearly, b = (R a )−1 La b. The element b is left bounded and b is right bounded,

bx =  (La )−1 e x = (La )−1 x ≤ (La )−1  x, ∀ x ∈ A0 .

xb  = x (R a )−1 e  = (R a )−1 x ≤ (R a )−1  x,

∀ x ∈ A0 .

4.2 The Spectrum of an Element

83

Let {xn } ⊂ A0 be a sequence converging to a.Then, by (v) of Lemma 4.2.3, we obtain

Lb a = lim (La )−1 e xn = lim (La )−1 xn = (La )−1 a = (La )−1 La e = e. n→∞

n→∞

Similarly one shows that R b a = e. Using Corollary 4.1.25, we get b = R b Lb a = Lb R b a = b. Hence, b ∈ Ab ∩ D(La ) ∩ D(R a ) and Lb a = R b a = e. This completes the proof.   We denote by Sb (A) the set of closable elements of A having a bounded inverse. Remark 4.2.6 In contrast with the case of Banach algebras, Sb (A) need not be an open subset of A. For instance, let us consider again the Banach quasi *-algebra (Lp (I ), C(I )) with I = [0, 1]. In this case, as seen in Example 4.1.12, (Lp (I ))b = L∞ (I ) and the norm  · b is exactly the L∞ -norm. Every neighborhood of the unit function u(x) = 1, for every x ∈ [0, 1], contains noninvertible elements: the function uε (x) = 0, for x ∈ [0, 2ε ) and uε (x) = 1, for x ∈ ( 2ε , 1] is in the ball B(u, ε) of Lp (I ), but it is not invertible, since it is zero on a set of positive measure. Proposition 4.2.7 Let (A[·], A0 ) be a normal Banach quasi *-algebra with unit e and a ∈ A an element having a bounded inverse a −1 . If b ∈ A and h := a − b ∈ Ab , with hb < a −1 −1 b , then b has a bounded inverse. Proof We have b = a • (e − a −1 • h). By the assumptions, the element e − a −1 • h is invertible in Ab , because a −1 • hb < 1 and Ab [ · b ] is a Banach algebra (Corollary 4.1.9(ii)). Hence, b has a bounded inverse.  

4.2.2 The Spectrum Let (A[ · ], A0 ) be a Banach quasi *-algebra with unit e. Definition 4.2.8 The resolvent ρ(a) of a ∈ A is the set   ρ(a) := λ ∈ C : a − λe ∈ Sb (A) . The set σ (a) := C \ ρ(a) is called the spectrum of a. Proposition 4.2.9 Let (A[ · ], A0 ) be a normal Banach quasi *-algebra with unit. Let a ∈ A, then the following statements hold: (i) the resolvent ρ(a) is an open subset of the complex plane; (ii) the resolvent function Rλ (a) := (a − λe)−1 ∈ Ab , λ ∈ ρ(a), is  · b -analytic on each connected component of ρ(a);

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4 Normed Quasi *-Algebras: Bounded Elements and Spectrum

(iii) for any two points λ, μ ∈ ρ(a), Rλ (a) and Rμ (a) commute and Rλ (a) − Rμ (a) = (μ − λ)Rμ (a) • Rλ (a). Proof (i) Let λ0 ∈ ρ(a) and λ ∈ C, such that |λ − λ0 | < Rλ0 (a)−1 b . Then, the series ∞ 

(λ0 − λ)n Rλ0 (a)n

n=1

converges to an element Sλ,a in Ab , with respect to  · b . Let now Tλ,a ≡ Rλ0 (a)(e + Sλ,a ). It is easily checked, using the  · convergence for the product Tλ,a (a − λe), that Tλ,a is a bounded inverse of a − λe. Hence, λ ∈ ρ(a). (ii) It follows immediately from the proof of (i). The proof of (iii) is straightforward.   The classical argument (see Theorem A.3.14) based on Liouville’s theorem can be applied to prove the following Proposition 4.2.10 Let a ∈ A. Then, σ (a) is non-empty. Example 4.2.11 Let us consider again the Banach quasi *-algebra (Lp (I ), C(I )) and let f ∈ Lp (I ). Then, it is easily seen that the spectrum σ (f ) of f coincides with its essential range; that is the set of all λ ∈ C, such that the set   x ∈ I : |f (x) − λ| < ε has positive Lebesgue measure, for every ε > 0.

 Definition 4.2.12 Let a ∈ A. The non-negative number r(a) = sup |λ|, λ ∈  σ (a) is called spectral radius of a. Remark 4.2.13 Of course, if a ∈ Ab , then σ (a) is the same set as that obtained regarding it as an element of the Banach *-algebra Ab . For an arbitrary element a ∈ A, σ (a), which is a closed set, could be an unbounded subset of C. The next proposition shows that, if a ∈ A \ Ab , then σ (a) is necessarily unbounded. Proposition 4.2.14 Let a ∈ A. Then, r(a) < ∞, if and only if, a ∈ Ab . Proof The “if” part has been discussed in the previous remark.

4.2 The Spectrum of an Element

85

Assume, now that r(a) < ∞. Then, the function λ → (a − λe)−1 is  · b analytic in the region |λ| > r(a). Therefore, it has there a  · b -convergent Laurent expansion (a − λe)−1 =

∞  xk , λk

|λ| > r(a),

k=1

with xk ∈ Ab , for each k ∈ N. As usually, 1 xk = 2π i

 γ

(a − λe)−1 dλ, λ−k+1

k ∈ N,

where γ is a circle centered in 0, with radius R > r(a). The integral on the right hand side converges with respect to  · b . The  · -continuity of the multiplication implies, as in the ordinary case, that axk =

1 2π i

 γ

a(a − λe)−1 1 dλ = 2π i λ−k+1

 γ

(a − λe)−1 dλ = xk+1 . λ−k

In particular, using Cauchy’s integral formula, we find ax1 = −a. This implies that   a ∈ Ab . Remark 4.2.15 If λ ∈ ρ(a) then all powers (a − λe)−n do exist in Ab , for every n ∈ N. This does not imply the existence of (a − λe)n , for n > 1. As an example, let us consider the Banach quasi *-algebra (L2 (I ), C(I )), where I = [0, 1] (cf. 1 Example 3.1.6). The function v(x) = x − 4 (we put v(0) = 0) is in L2 (I ); obviously, 1 n 0 ∈ ρ(v), since v −1 (x) = x 4 ∈ L∞ (I ). We have v −n (x) = x 4 ∈ L2 (I ), for all 1 n ∈ N, but v 2 (x) = x − 2 ∈ L2 (I ).

4.2.3 The *-Semisimple Case Proposition 4.2.16 Let (A[ · ], A0 ) be a *-semisimple Banach quasi *-algebra. Then, (A[ · ], A0 ) is normal. Proof Assume that a, b ∈ Ab . Then, for every ϕ ∈ SA0 (A) and z ∈ A0 , we have







ϕ (a b)z, z = ϕ (La b)z, z = ϕ (R b a)z, z = ϕ (a b)z, z . Therefore,

ϕ (a b − a b)z, z = 0,

∀ ϕ ∈ SA0 (A), z ∈ A0 .

Hence, by *-semisimplicity a b = a b, for all a, b in Ab . This completes the proof.  

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4 Normed Quasi *-Algebras: Bounded Elements and Spectrum

Definition 4.2.17 A Banach quasi *-algebra (A[ · ], A0 ) is called quasi regular, respectively regular, if it is *-semisimple and   a = max p(a), p(a ∗ ) , resp., p(a) = a,

∀ a ∈ A.

We recall that the seminorm p was defined in (3.1.9) by p(a) :=

sup

ϕ∈SA0 (A)

ϕ(a, a)1/2 , a ∈ A.

It is clear that every regular Banach quasi *-algebra is quasi regular. Moreover, if (A[ · ], A0 ) is commutative and quasi regular, then it is regular. In fact, in this case, if ϕ ∈ SA0 (A) then ϕ ∗ ∈ SA0 (A), where ϕ ∗ (a, b) = ϕ(b∗ , a ∗ ), a, b ∈ A. This easily implies that p(a) = p(a ∗ ), for every a ∈ A. Examples will be given after Proposition 4.2.24. Remark 4.2.18 Taking into account Proposition 3.1.18, if (A[ · ], A0 ) is a regular Banach quasi *-algebra, then for each element a ∈ A, there exists ϕ ∈ SA0 (A), such that ϕ(a, a) = a2 , a property, which is closely reminiscent of the behaviour of unital C*-algebras (see, Proposition A.5.4). It is not expected that every *-semisimple Banach quasi *-algebra (A[ · ], A0 ) is regular, because this is not true in the case, where A = A0 and  ·  =  · 0 , that is when A[·] is a *-semisimple Banach *-algebra. In that situation, the C*-seminorm p defined in Proposition A.5.1, is a norm which, in general, does not coincide with the initial norm  · 0 . Every *-semisimple Banach *-algebra is, in fact, an A*algebra, but not necessarily a C*-algebra. A well-known example is provided by the convolution algebra L1 (R), which is an A*-algebra but not a C*-algebra (see [21, Corollary (4.6.10); Example A.3.1]). Our next goal is to show that, for regular Banach quasi *-algebras the set of elements having finite spectral radius can also be described in terms of the seminorm q defined in (3.1.13). We begin with the following Proposition 4.2.19 Let (A[ · ], A0 ) be a *-semisimple Banach quasi *-algebra. Then, for every a ∈ A, the maps La : x ∈ A0 → ax ∈ A,

Ra : x ∈ A0 → xa ∈ A

are closable in A. Proof Let a ∈ A and {xn } a sequence in A0 ,  · -converging to zero and such that axn → b, with respect to  · . Then, if ϕ ∈ SA0 (A) and y1 , y2 ∈ A0 , we get

|ϕ(by1 , y2 )| ≤ |ϕ (b − axn )y1 , y2 | + |ϕ(xn y1 , a ∗ y2 )| ≤ b − axn  y1 0 y2 0 + xn  y1 0 a ∗ y2  → 0.

4.2 The Spectrum of an Element

87

Therefore, ϕ(by1 , y2 ) = 0, for every ϕ ∈ SA0 (A) and for every y1 , y2 ∈ A0 . From this we obtain that ϕ(b, b) = 0, for every ϕ ∈ SA0 (A). Therefore, b = 0, i.e., La is closable (see Proposition B.4.5 in Appendix B). The proof for Ra is similar.   As we have seen, it is possible to introduce in a *-semisimple Banach quasi *-algebra (A[ · ], A0 ) a weak multiplication  which makes it into a partial *algebra. As a consequence of this, for every a ∈ A, the operators La , Ra have closed extensions (see Proposition 3.1.34) defined, respectively, in Rw (a), Lw (a) a , R a . It is clear that (for the notation of the left parts and denoted, respectively by L a and R a ⊂ R a . of the inclusions, see beginning of Sect. 4.1.1) La ⊂ L The following proposition gives an interesting variant of the associativity law. Proposition 4.2.20 Let (A[ · ], A0 ) be a *-semisimple Banach quasi *-algebra. Suppose that a, b, c are elements of A, such that (a b)c is well defined. If c ∈ D(Lb ), then a (bc) is well defined and (a b)c = a (bc). Proof Since c ∈ D(Lb ), there exists a sequence {zn } ⊂ A0 such that c − zn  → 0 and Lb c − bzn  → 0 (see discussion before Lemma 4.2.3). Then, for every ϕ ∈ SA0 (A) and x, y ∈ A0 , we have





ϕ (a b)c x, y = ϕ cx, (a b)∗ y = ϕ cx, (b∗ a ∗ )y = lim ϕ(zn x, (b∗ a ∗ )y) = lim ϕ(b(zn x), a ∗ y) n→∞

n→∞





= lim ϕ (bzn )x, a ∗ y = ϕ (Lb c)x, a ∗ y n→∞

= ϕ (bc)x, a ∗ y . From these equalities it follows that a (bc) is well defined and the required equality, indeed holds.   The next Proposition 4.2.21 provides a handy criterion for the existence of a bounded inverse of an element. Proposition 4.2.21 Let (A[ · ], A0 ) be a *-semisimple Banach quasi *-algebra with unit e. Then, a ∈ A has a bounded inverse, if and only if, both of the following conditions hold: (i) there exists γ > 0, such that   min ax, xa ≥ γ x,

∀ x ∈ A0 ;

    (ii) the sets ax, x ∈ A0 and xa, x ∈ A0 are both dense in A.

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4 Normed Quasi *-Algebras: Bounded Elements and Spectrum

Proof Suppose that (i) and (ii) hold. Using the definitions of La and R a , it is easy to prove that (i) extends as follows: La y ≥ γ y,

∀ y ∈ D(La ) and R a z ≥ γ z,

∀ z ∈ D(R a ).

(4.2.7)

Hence, both La and R a are injective maps and their ranges, La D(La ), R a D(R a ), are dense by assumption. The inequalities (4.2.7) imply also that La D(La ) and R a D(R a ) are closed in A. Indeed, let w = limn→∞ wn with wn ∈ La D(La ). Then, for every n ∈ N there is a unique yn ∈ D(La ) such that wn = La yn . It follows from (4.2.7) that the sequence {yn } is Cauchy. Then, it converges to an element y ∈ A. The closedness of La implies that y ∈ D(La ) and La y = limn→∞ La yn = w. This proves the closedness of La D(La ). Since it is also dense, it follows that La D(La ) = A. Similarly, one shows that R a D(R a ) = A. Hence, both La and R a have bounded inverses, (La )−1 and (R a )−1 , everywhere defined in A. Taking into account Propositions 4.2.16 and 4.2.5, we conclude that a has a bounded inverse. The converse is easily seen.   We consider now the seminorm q defined in (3.1.13). As we have seen, in general, this seminorm is defined on a domain D(q), which is smaller than A and it has the following properties: (a) q(a ∗ ) = q(a), ∀ a ∈ A; (b) q(x ∗ x) = q(x)2 , ∀ x ∈ A0 ; i.e., it is an extended C*-seminorm, in the sense of [77]. If (A, A0 ) has a unit e and SA0 (A) is sufficient, then q is a C*-norm. One has p(a) ≤ q(a),

∀ a ∈ D(q).

(4.2.8)

The space D(q) endowed with the topology defined by q is denoted by Aq . Then, we have the following Proposition 4.2.22 Let (A[ · ], A0 ) be a *-semisimple Banach quasi *-algebra with a unit e. Then, Aq is a normed space containing A0 as a subspace. Moreover, if (A[ · ], A0 ) is quasi regular, then Aq is a Banach space. Proof The first part of the statement follows from (i) of Proposition 3.1.40. To prove that, Aq is a Banach space, when (A[·], A0 ) is quasi regular, we only have to show its completeness. Let {an } be a q-Cauchy sequence in Aq . The inequality (4.2.8), in the quasi regular case, becomes a ≤ q(a), for all a ∈ Aq . Therefore, {an } is also  · -Cauchy. Using the  · -completeness of A, we conclude that there exists an element a ∈ A, which is the  · -limit of an . Let ϕ ∈ PA0 (A). Then, ϕ(a, a) = limn→∞ ϕ(an , an ). The sequence q(an ) is bounded, because {an } is q-Cauchy. If M is its supremum, we have ϕ(an , an )1/2 ≤ q(an )ϕ(e, e) ≤ Mϕ(e, e).

4.2 The Spectrum of an Element

89

In conclusion, ϕ(a, a)1/2 ≤ Mϕ(e, e). Thus, clearly q(a) < ∞, i.e., a ∈ Aq . Finally, using the uniqueness of the limit in the completion of Aq , we conclude that a = q − limn→∞ an . Therefore, Aq is complete.   We observe that, in general, the inclusion A0 ⊆ Aq is proper. For instance, in (Lp (I ), C(I )) any step function s defined on [0, 1] belongs to Lp (I ), but not to C(I ). It is immediate to verify that s ∈ (Lp (I ))q . Proposition 4.2.23 Let (A[·], A0 ) be a regular Banach quasi *-algebra with unit e. Let a ∈ Aq and λ ∈ C, such that |λ| > q(a). Then, a − λe has a bounded inverse (a − λe)−1 in Ab . Thus, 

 λ ∈ C : |λ| > q(a) ⊆ ρ(a).

Proof If a ∈ D(q), by our assumption and definition of q(a), one has that |λ| > q(a) ≥ ϕ(ay, ay),

∀ y ∈ A0 , such that ϕ(y, y) = 1.

Therefore, for every x ∈ A0 ,

1/2

1/2 ≥ |λ|ϕ(x, x) − ϕ(ax, ax) ϕ (a − λe)x, (a − λe)x

1/2 ≥ |λ|ϕ(x, x) − q(a) . Taking the supremum over the family SA0 (A) we obtain



p (a − λe)x ≥ p(x) |λ| − q(a) ,

∀ x ∈ A0 .

Taking into account the regularity of (A[ · ], A0 ), we finally obtain (a − λe)x ≥ x(|λ| − q(a)),

∀ x ∈ A0 .

(4.2.9)

Now we prove that if q(a) < ∞ and |λ| > q(a) the sets   Ran (La−λe ) := (a − λe)y : y ∈ A0 ,

  Ran (Ra−λe ) := y(a − λe) : y ∈ A0

are  · -dense in A. If it is not so, there would exist a non zero ·-continuous functional f on A, such that f ((a−λe)y) = 0, for every y ∈ A0 . Therefore, we should have f (ay) = λf (y), for every y ∈ A0 . From the  · -continuity of f we obtain |f (ay)| ≤ f  ay, for every y ∈ A0 , where f  is the norm of f in the dual Banach space A of A[ · ].

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4 Normed Quasi *-Algebras: Bounded Elements and Spectrum

From Proposition 3.1.40(iii), we obtain p(ay) ≤ q(a)p(y), for all a ∈ D(q) and y ∈ A0 . Thus, by the regularity of (A[ · ], A0 ), |f (ay)| ≤ f  ay = f  p(ay) ≤ f  q(a)p(y) = f  q(a)y,

(4.2.10)

for all y ∈ A0 . The functional fa , defined by fa (y) := f (ay), y ∈ A0 , is  · continuous, since |fa (y)| = |λf (y) ≤ |λ| f  y,

∀ y ∈ A0 .

(4.2.11)

An easy computation shows that fa  = |λ|f  . From this and the inequalities (4.2.10), (4.2.11) we deduce |λ| ≤ q(a), which is a contradiction. A similar argument shows the corresponding statement for Ran (Ra−λe ). By Proposition 4.2.21 it finally follows that a − λe has a bounded inverse   Let (A[ · ], A0 ) be *-semisimple normed quasi *-algebra. Then, the seminorm p on A becomes a norm as follows from the very definitions. Denote by AS the completion of A0 [p]. In this regard, we have the following Proposition 4.2.24 Let (A[ · ], A0 ) be a *-semisimple Banach quasi *-algebra and assume that p(a ∗ ) = p(a), for every a ∈ A. Then, there exists a regular Banach quasi *-algebra (AS , A0 ), such that AS contains A, as a dense subspace. Proof Note that (AS , A0 ) is a Banach quasi *-algebra, because of Proposition 3.1.40(iv) and the assumption that p(a ∗ ) = p(a), for every a ∈ A. We now prove that A can be identified with a subspace of AS . Indeed, if a ∈ A then there exists a sequence {xn } ⊂ A0 , such that xn − a → 0. n→∞

Since p(a) ≤ a, for every a ∈ A, {xn } is a Cauchy sequence with respect to p, too. Thus, there exists an element a ∈ AS , such that p(xn − a) → 0. n→∞

The element a does not depend on the particular sequence {xn } used to approximate a in A. Indeed, if {xn } is another sequence with the same property, then p(xn − xn ) ≤ xn − xn  → 0,

as n → ∞.

We have defined in this way a map i : a ∈ A → a ∈ AS ; we shall prove that i is injective.

4.2 The Spectrum of an Element

91

Assume that a = 0, for some a ∈ A and let {xn } be a sequence in A0 approximating a in the norm of A and such that p(xn ) → 0; this implies that ϕ(xn , xn ) → 0, for each ϕ ∈ SA0 (A). Therefore, ϕ(a, a) = lim ϕ(xn , xn ) = 0. n→∞

From the sufficiency of SA0 (A), we obtain a = 0 (see Definition 3.1.17 and Corollary 3.1.24). To conclude the proof, we need to show that SA0 (AS ) is sufficient and that (AS , A0 ) is regular. First, we prove that the two families of sesquilinear forms SA0 (A) and SA0 (AS ) can be identified. Indeed, assume that ∈ SA0 (AS ), then its restriction A to A belongs, as it is easily seen, to SA0 (A). Conversely, if 0 ∈ SA0 (A), making use of the Cauchy–Schwarz inequality, we obtain | 0 (a, b) ≤ p(a)p(b),

∀ a, b ∈ A.

Therefore, 0 has a unique extension to AS and ∈ SA0 (AS ). Taking this fact into account, the sufficiency of SA0 (AS ) follows by the definition of AS . Thus, (AS , A0 ) is *-semisimple and the extension, say pS , of p on AS is also a norm. The regularity is now a simple consequence of the definition of pS .   Example 4.2.25 The BQ*-algebra (see Definition 2.1.4) (Lp (I ), C(I )) is regular [50], if and only if, p ≥ 2. For 1 ≤ p < 2, SC(I ) (Lp (I )) = {0}. In the case of the non-commutative Lp in Example 3.1.7, it has been proved in [52] that, for finite , (Lp (), M) is regular if p ≥ 2. Example 4.2.26 The Banach quasi *-algebra (H, A0 ) of Example 3.1.8 is *semisimple, since SA0 (A) contains the inner product ·|·. For the same reason, (H, A0 ) is regular. Finally, using the uniqueness of the limit in the completion of Aq , we conclude that a = q − limn→∞ an . Therefore, Aq is complete. We can now prove the following, where for the term A*-algebra, the reader is referred to [21, p. 181]. Theorem 4.2.27 Let (A[·], A0 ) be a regular Banach quasi *-algebra with unit e. Then, D(q) coincides with the set Ab of all bounded elements of A. Moreover, q(a) = ab ,

∀ a ∈ Ab .

Therefore, Ab [ · b ] is a C*-algebra. Proof Proposition 4.2.23 shows that D(q) ⊆ Ab . On the other hand, let us consider, for each ϕ ∈ PA0 (A), the linear functional ωϕ defined by ωϕ (a) := ϕ(a, e), a ∈ Ab .

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4 Normed Quasi *-Algebras: Bounded Elements and Spectrum

A simple limit argument shows that ωϕ is positive (i.e., ω(a ∗ • a) ≥ 0, for each a ∈ Ab ), so the corresponding GNS representation πϕ is bounded and πϕ (a) ≤ ab . Then, if a ∈ Ab ,  q(a) = sup ϕ(ax, ax)1/2 : ϕ ∈ PA0 (A), x ∈ A0 , ϕ(x, x) = 1}  = sup πϕ (a) ≤ ab . ϕ∈PA0 (A)

On the other hand, from Proposition 3.1.40(iii) follows that xa = p(xa) ≤ q(a)p(x) = q(a)x,

∀ a ∈ D(q), x ∈ A0 .

This implies that ab ≤ q(a), for every a in D(q) (see (4.1.2)). Thus, in conclusion,    · b is a C*-norm. A further characterization of the set of bounded elements of (A[ · ], A0 ), in the case where SA0 (A) is sufficient, can be obtained in terms of *-representations. In particular, under the latter assumption, (A[ · ], A0 ) accepts a faithful *representation. Theorem 4.2.28 Let (A[ · ], A0 ) be a *-semisimple Banach quasi *-algebra with unit e. Then, (A[ · ], A0 ) admits a faithful *-representation π in a Hilbert space H. Moreover,   Ab = a ∈ A : π(a) ∈ B(H) and π(a) = q(a),

∀ a ∈ Ab .

Proof Let ϕ ∈ SA0 (A) and let πϕ be the corresponding GNS representation with dense domain Dϕ ⊆ Hϕ . Put H=

 ϕ∈SA0 (A)

 Hϕ = (ξϕ )ϕ∈SA

0

(A)



: ξϕ ∈ H ϕ ,

 ξϕ  < ∞ , 2

ϕ∈SA0 (A)

with the usual inner product (ξϕ )|(ηϕ ) =



ξϕ |ηϕ ,

(ξϕ ), (ηϕ ) ∈ H.

ϕ∈SA0 (A)

Let  D = (ξϕ )ϕ∈SA

0

(A)

∈ H : ξϕ ∈ Dϕ , ϕ ∈ SA0 (A) :



ϕ∈SA0 (A)

 πϕ (a)ξϕ  < ∞, ∀ a ∈ A . 2

4.2 The Spectrum of an Element

93

Then, D is a dense domain in H and so we can define π(a)(ξϕ ) = (πϕ (a)ξϕ ),

∀ a ∈ A, (ξϕ )ϕ∈SA

0

(A)

∈ D.

Thus, π(a) ∈ L† (D, H), for each a ∈ A and π : a ∈ A → π(a) ∈ L† (D, H) is a *-representation of (A[ · ], A0 ). Moreover, π is faithful, since π(a) = 0 ⇔ πϕ (a) = 0, ∀ ϕ ∈ SA0 (A) ⇔ ϕ(a, a) = 0, ∀ ϕ ∈ SA0 (A). The sufficiency of SA0 (A) (see Corollary 3.1.24) now implies that a = 0. Finally, π(a) is bounded, if and only if, each πϕ , ϕ ∈ SA0 (A), is bounded and sup

ϕ∈SA0 (A)

πϕ (a) < ∞;

in this case, π(a) =

sup

ϕ∈SA0 (A)

πϕ (a) =

sup

ϕ∈PA0 (A)

πϕ (a),

a ∈ A.

The latter equality follows from the fact that for every ϕ ∈ PA0 (A), there exists ψ ∈ SA0 (A), such that πϕ (a) = πψ (a), a in A. For this it is enough to choose ϕ , ϕ as before, where n(ϕ)2 = supa∈A ϕ(a,a) . Then, PA0 (A) = SA0 (A), ψ = n(ϕ) a2 therefore we obtain sup ϕ∈PA0 (A)

πϕ (a) = sup{ϕ(ax, ax)1/2 : ϕ ∈ PA0 (A), x ∈ A0 , ϕ(x, x) = 1} = q(a).

Hence, π(a) = q(a), for every a ∈ A and this concludes the proof.

 

If (A[ · ], A0 ) has a unit, the *-representation π constructed in Theorem 4.2.28 enjoys the following property: for every ϕ ∈ SA0 (A), there exists a vector ξ ∈ D, with ξ  = 1, such that ϕ(a, b) = π(a)ξ |π(b)ξ ,

∀ a, b ∈ A.

Borrowing the terminology from the theory of *-algebras (see, e.g., [7, Definition 3.1.16] and also Remark A.6.16), we will call a *-representation π with this property universal. Then, we can restate Theorem 4.2.28 in the following terms, having thus a Gelfand–Naimark type theorem for *-semisimple Banach quasi *-algebras with unit.

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4 Normed Quasi *-Algebras: Bounded Elements and Spectrum

Theorem 4.2.29 Let (A[ · ], A0 ) be a Banach quasi *-algebra with unit. The following statements are equivalent: (i) there exists a faithful universal *-representation π of (A[ · ], A0 ); (ii) (A[ · ], A0 ) is *-semisimple.

Chapter 5

CQ*-Algebras

This chapter is devoted to a special class of Banach quasi *-algebras, the so-called CQ*-algebras. Their essential feature consists of the fact that they contain a C*algebra as dense subspace.

5.1 Basic Aspects Definition 5.1.1 Let (A[ · ], A0 ) be a Banach quasi *-algebra. We say that (A[ · , A0 ) is a proper CQ*-algebra if (i) A0 is a C*-algebra with norm  · 0 and involution * inherited by that of A; (ii) A0 is dense in A[ · ]; (iii) x0 = sup ax, x ∈ A0 . a∈A,a≤1

We have defined the norm  · 0 for a Banach quasi *-algebra (A[ · , A0 ) by (3.1.4) (see also (3.1.1)). It is worth mentioning that condition (iii) of Definition 5.1.1, which is seemingly imposing a stronger requirement, is exactly equivalent to the definition given in Chap. 3, due to the fact that A0 is supposed to be a C*-algebra. This is a consequence of the following lemma which can be applied, for x ∈ A0 , to x0 as defined in (3.1.4) and to x1 := sup ax. a∈A,a≤1

Lemma 5.1.2 Let A0 [ · 0 ] be a C*-algebra and  · 1 another norm on A0 , which makes of it a normed algebra. Suppose that x1 ≤ x0 , for every x ∈ A0 . Then, x1 = x0 ,

∀ x ∈ A0 .

Proof Let x = x ∗ ∈ A0 and let M(x) denote the abelian C*-algebra generated by x. Since every norm that makes an abelian C*-algebra into a normed algebra is © Springer Nature Switzerland AG 2020 M. Fragoulopoulou, C. Trapani, Locally Convex Quasi *-Algebras and their Representations, Lecture Notes in Mathematics 2257, https://doi.org/10.1007/978-3-030-37705-2_5

95

96

5 CQ*-Algebras

necessarily stronger than the C*-norm [22, Theorem 1.2.4], we obtain the equality x1 = x0 , ∀ x = x ∗ ∈ A0 . For an arbitrary element y ∈ A0 , we have y20 = y ∗ y0 = y ∗ y1 ≤ y ∗ 1 y1 . But y ∗ 1 ≤ y ∗ 0 and so y20 ≤ y0 y1 and this implies that y0 ≤ y1 .   A proper CQ*-algebra can be viewed as the completion of an arbitrary C*algebra A0 [ · ]0 with respect to a weaker norm. Indeed, we have Proposition 5.1.3 Let A0 be a C*-algebra, with norm  · 0 and involution *. Let  ·  be another norm on A0 , weaker than  · 0 , i.e., x ≤ x0 ,

∀ x ∈ A0

and such that the following conditions are satisfied: (i) xy ≤ x y0 , ∀ x, y ∈ A0 ; (ii) x ∗  = x, ∀ x ∈ A0 . Let A denotes the  · -completion of A0 . Then, (A[ · ], A0 ) is a proper CQ*algebra. Proof Let A be the Banach space completion of A0 [ · ]. For x ∈ A0 , put x1 := sup ax, a ∈ A, a≤1

(5.1.1)

It follows that the operator Rx : A[ · ] → A[ · ], such that Rx (a) := ax, for all a ∈ A, is bounded and xy1 ≤ x1 y1 , for every x, y ∈ A0 . Moreover, from (i), we obtain x1 ≤ x0 ,

∀ x ∈ A0 .

Hence, the statement follows by Lemma 5.1.2 and Definition 5.1.1.

 

Corollary 5.1.4 Let (A[ · ], A0 ) be a proper CQ*-algebra and B0 any *subalgebra of A0 . Let M0 (B0 ) be the closure of B0 in A0 and M(B0 ) the closure of B0 in A. Then, (M(B0 ), M0 (B0 )) is a proper CQ*-algebra.

5.1.1 Commutative CQ*-Algebras We begin with giving some examples of commutative CQ*-algebras. Example 5.1.5 (CQ*-Algebras of Functions) Let μ be a measure in a non-empty point set X. Let M + be the collection of all μ-measurable nonnegative functions on

5.1 Basic Aspects

97

X. We assume that to each f ∈ M + it corresponds a number ρ(f ) ∈ [0, ∞], such that: (i) (ii) (iii) (iv)

ρ(f ) = 0, if and only if, f = 0, a.e. in X; ρ(f1 + f2 ) ≤ ρ(f1 ) + ρ(f2 ), ∀ f1 , f2 ∈ M + ; ρ(λf ) = λρ(f ), ∀ λ ∈ R+ , f ∈ M + ; let {fn } ⊂ M + and fn ↑ f a.e. in X. Then, ρ(fn ) ↑ ρ(f ).

Following [30] we call ρ a function norm. Let us define   Lρ := f ∈ M + : ρ(f ) < ∞ . With this definition it has been proved in [30] that the space Lρ is a Banach space, that is complete, with respect to the norm f  ≡ ρ(|f |). Some Lρ spaces generate examples of abelian proper CQ*-algebras. (A) Let (X, μ) be a measure space with μ a regular Borel measure on a compact Hausdorff space X. As usual, we denote by Lp (X, μ) the Banach space of all (equivalence classes of) measurable functions f : X −→ C, such that  f p :=

!1/p |f |p dμ < ∞.

X

On Lp (X, μ) we consider the natural involution f ∈ Lp (X, μ) → f ∗ ∈ Lp (X, μ) with f ∗ (x) = f (x). Clearly Lp (X, μ) is an Lρ space (with  ·  ≡  · p ). We denote by C(X) the C*-algebra of continuous functions defined on X. The pair (Lp (X, μ), C(X)) provides the basic commutative example of a Banach quasi *-algebra. It turns also out that (Lp (X, μ), C(X)) is a proper abelian CQ*-algebra, for any p ≥ 1, since the p-norm satisfies all the conditions of Proposition 5.1.3. These spaces have been analyzed with a certain care in [50]. (B) Let X be a compact Hausdorff space and M = {μα , α ∈ I } a family of Borel measures on X, for which there exists a constant c > 0, such that μα (X) ≤ c, for all α ∈ I . Let  · p,α be the norm on Lp (X, μα ). Of course, each norm is related to a particular function norm ρp,α (·). Let us define, for f ∈ C(X), the function f p,I := sup f p,α . α∈I

In [30] it is shown that the map ρp,I related to this norm still satisfies all the p requirements of a function norm, so that the completion LI (X, M) of C(X) with respect to  · p,I , is a Banach space. The norm  · p,I also satisfies the conditions of Proposition 5.1.3. Indeed, since f p,I ≤ cf ∞ , for all

98

5 CQ*-Algebras

f ∈ C(X), it follows that  · p,I is finite on C(X) and really defines a norm on C(X) satisfying (i) f ∗  = f  , ∀ f ∈ C(X) ; p,I

p,I

(ii) f gp,I ≤ f p,I g∞ ,

∀ f, g ∈ C(X).

p

Therefore, (LI (X, M), C(X)) is a commutative proper CQ*-algebra. It is p clear that LI (X, M) can be identified with a subspace of Lp (X, μα ), for all p α ∈ I . It is obvious that LI (X, M) may contain also noncontinuous functions (depending on the set X and on the family M of measures). (C) Let X, M and ρp,α be as above. For a set {cα } of positive constants, we define ρp (f ) :=



cα ρp,α (f ).

α∈I

Then, the space Lp (X, M) (the completion of C(X) with respect to the norm p  · p generated by ρp ) is a Banach space, which contains the space LI (X, M) of the previous example, if the set of numbers {cα } is summable. Again, (Lp (X, M), C(X)) is an abelian proper CQ*-algebra. (D) Interesting examples of commutative proper CQ*-algebras are also provided by Nachbin spaces (see [43]). Let X be a locally compact Hausdorff space and v an upper semicontinuous nonnegative function, such that inft∈X v(t) > 0. As usual, C(X) and C0 (X) stand, respectively, for the set of continuous functions on X and for the set of continuous functions on X vanishing at infinity. Let us consider the spaces   Cbv (X) := f ∈ C(X) : vf is bounded on X and   C0v (X) := f ∈ C(X) : vf vanishes at infinity on X . The weighted uniform topology is defined by the norm f v := sup v(t)|f (t)|, f ∈ Cbv (X). t∈X

It turns out that both Cbv (X) and C0v (X), when equipped with the weighted uniform topology, are Banach modules over C0 (X). Actually, (Cbv (X), C0 (X)) and (C0v (X), C0 (X)) are commutative proper CQ*-algebras. In this subsection we shall describe the structure of commutative CQ*-algebras. As is known, for C*-algebras the situation is completely clear: a commutative C*-algebra with unit is isometrically *-isomorphic to the C*-algebra C(X) of all Cvalued continuous functions on the compact space X of all characters of C(X). The respective correspondence in this case is the so-called Gelfand map. CQ*-algebras

5.1 Basic Aspects

99

do not behave so nicely: the first reason is that Proposition 5.1.3 allows the existence of non isomorphic CQ*-algebras over C(X); indeed, in the case considered in Example 5.1.5 (A), it is clear that Lp (X) is not isomorphic to Lq (X), if p = q, in a usual situation; the second reason is that, as is known [4, 8, 18] already for Banach *-algebras the Gelfand map is not, in general, an isometric *-isomorphism. Proposition 5.1.6 Let (A[ · ], A0 ) be a commutative proper CQ*-algebra and X the space of all characters of A0 . Then, the following statements hold: (i) if A0 admits a faithful state ω, which is continuous with respect to the norm  ·  of A, then there exists a linear map φ from A into C(X) (the dual of the Banach space C(X)), whose restriction to A0 is a linear isomorphism of A0 onto C(X); (ii) if, in addition, for some positive constant γ > 0 |ω(y ∗ x)| ≤ γ xy,

∀ x, y ∈ A0 ,

then there exists a regular Baire measure μ on X and a linear map φ from A into L2 (X, dμ), whose restriction to A0 is a linear isomorphism of A0 onto C(X). In both cases φ preserves involution. Proof (i) Let ω be a faithful state, which is continuous with respect to the norm  ·  of A; then ω has a continuous extension (denoted by the same symbol) to the whole of A; for each a ∈ A, the linear functional ωa defined by ωa (x) := ω(ax), x ∈ A0 , is therefore, bounded on A0 . Let us now consider the linear functional  ωa on C(X) defined by  ωa ( x ) := ωa (x), x ∈ A0 and let f : a ∈ A → f (a) :=  ωa ∈ C(X) . It is easily shown that f is a linear map of A into C(X) . We define φ := f A0 . Then, φ is a linear isomorphism: indeed, if x ∈ A0 and  ωx ( y ) = 0, for every y ∈ A0 , then in particular,  ωx ( x ∗ ) = ω(xx ∗ ) = 0 and then, by the faithfulness of ω, x = 0. (ii) Set ϕ0 (x, y) := ω(y ∗ x), x, y ∈ A0 . By the assumption for ω in (ii), ϕ0 can be extended, by continuity, to a positive sesquilinear form ϕ on A × A. Since ω is linear and continuous on A0 ,  ω (defined exactly as  ωa , above) is continuous on C(X), so that by the Riesz representation theorem [14], there

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exists a unique Borel measure μω on X, such that   x (ρ)dμω (ρ),

ω(x) =  ω( x) =

∀ x ∈ A0 .

X

 

For a ∈ A the conjugate linear form Fa on C(X) defined by Fa ( x ) = ϕ(a, x), x ∈ A0 , is bounded in L2 (X, dμω ), due to the Schwarz inequality. Therefore, there exists a function  a ∈ L2 (X, dμω ), such that  x) =  a (ρ) x (ρ)dμω (ρ), x ∈ A0 . Fa ( X

The map φ : a ∈ A → φ(a) =  a ∈ L2 (X, dμω ) is, as it is readily checked, a linear map of A into L2 (X, dμω ), preserving the involution ∗ of A0 . It is easy to see that φ satisfies the requirements of our proposition. The fact that φ A0 is a linear isomorphism of A0 onto C(X) follows, as before, from the faithfulness of ω. Remark 5.1.7 It is clear that the assumption |ω(y ∗ x)| ≤ γ xy, for all x, y ∈ A0 , made in Proposition 5.1.6(ii), implies the continuity of ω required in (i). The converse is, however, not true in general. This could appear in contradiction to the fact that any separately continuous sesquilinear form on a Banach space, as A, is necessarily jointly continuous. But, as a matter of fact, the continuity of ω does not imply the separate continuity of ϕ0 . There is, however, one relevant exception: if the state ω of Proposition 5.1.6(i) can be taken to be pure, then it is multiplicative, therefore ϕ0 (x, y) = ω(x)ω(y),

∀ x, y ∈ A0 .

Hence, ϕ0 is separately continuous with respect to the norm  ·  of A and the same holds for its extension ϕ to the whole A. Now we show that the *-semisimple and commutative case is completely understood: in fact, as we shall see below, any CQ*-algebra with these two properties can be thought as a CQ*-algebra of functions. We remind the reader that, in the case of the Lp -spaces, *-semisimplicity occurs, if and only if, p ≥ 2 (see Example 3.1.29). Let X be a compact Hausdorff space and M = {μα : α ∈ I} a family of Borel measures on X. p The CQ*-algebra (LI (X, M), C(X)) constructed in Example 5.1.5(B) is *-semisimple, for p ≥ 2. This depends on the fact that, for each α, an element p of SC(X) (Lp (X, μα )) gives rise, by restriction, to an element of SC(X) (LI (X, M)) (for the latter notation, see Definition 3.1.14). Proposition 5.1.8 Let (A[ · ], A0 ) be a *-semisimple commutative CQ*-algebra with unit e. Then, there exists a family M of Borel measures on the Hausdorff

5.1 Basic Aspects

101

compact space X of the characters of A0 and a map : a ∈ A → (a) ≡  a ∈ L2I (X, M) with the following properties: (i) (ii) (iii) (iv)

extends the Gelfand transform of elements of A0 and (A) ⊃ C(X);

is linear and injective;

(ax) = (a) (x), ∀ a ∈ A, x ∈ A0 ;

(a ∗ ) = (a)∗ , ∀ a ∈ A.

Thus, A can be identified with a subspace of L2I (X, M). If A is regular (cf. Definition 4.2.17), that is, if a2 =

sup

ϕ∈SA0 (A)

ϕ(a, a), a ∈ A ,

then is an isometric *-isomorphism of A onto L2I (X, M). Proof Define first on A0 as the usual Gelfand map x ∈ C(X).

: x ∈ A0 →  As it is well-known, the Gelfand map is an isometric *-isomorphism of A0 onto C(X). Let ϕ ∈ SA0 (A). Define the linear functional ω on C(X) by ω( x ) := ϕ(x, e), x ∈ A0 . It is easy to check that ω is bounded on C(X); then, by the Riesz representation theorem, there exists a unique regular positive Borel measure μϕ on X, such that  ω( x ) = ϕ(x, e) =

 x (η)dμϕ (η),

∀ x ∈ A0 .

X

We have μϕ (X) ≤ e2 , for all ϕ ∈ SA0 (A). Let M ≡ {μϕ : ϕ ∈ SA0 (A)} and let L2SA (A) (X, M) be the CQ*-algebra 0 constructed as above. Now, if a ∈ A, there exists a sequence {xn } ⊂ A0 converging to a in the norm of A. We have then 2 xn − x" m 2,SA

0

(A)

=

sup

ϕ∈SA0 (A)

ϕ(xn − xm , xn − xm ) ≤ xn − xm 2 → 0 .

Let  a be the  · 2,SA0 (A) -limit of {xn } in L2SA

0

(A) (X, M).

Define

(a) :=  a , a ∈ A. Evidently,  a 2,SA0 (A) = supϕ∈SA (A) ϕ(a, a), a ∈ A. This implies that if  a = 0, 0 then ϕ(a, a) = 0, for all ϕ ∈ SA0 (A) and thus a = 0, since A is *-semisimple. The proof of (ii), (iii) and (iv) is straightforward.

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Now, if A is regular, it follows immediately from the preceding discussion that

is an isometry. We conclude by proving, in this case, that is onto. Let h be an element of L2SA (A) (X, M). Then, there exists a sequence { an } ⊂ C(X) converging 0 to h with respect to  · 2,SA0 (A) . The sequence {an } converges to some a ∈ A. It is easily seen that h =  a . Hence, is onto.  

5.2 General CQ*-Algebras In the previous sections we have considered proper CQ*-algebras, only in the commutative case. However, there are also noncommutative examples, where just here we give an easy one. Other examples are provided by noncommutative Lp spaces in Sect. 5.6.2. Example 5.2.1 Let A0 [ · 0 ] be a C*-algebra with unit e. Let be a linear map of A0 into itself with (x ∗ ) = (x)∗ , for every x ∈ A0 . Suppose that the following inequality is fulfilled  (xy)0 ≤  (x)0 y0 ,

∀ x, y ∈ A0 .

(5.2.2)

Let us assume that  (e)0 = 1 and define a new norm on A0 by x :=  (x)0 ,

x ∈ A0 .

It is easy to verify that this norm satisfies the conditions of Proposition 5.1.3. Therefore, the  · -completion A of A0 is a proper CQ*-algebra over A0 . Of course, the inequality (5.2.2) automatically holds if is a *-homomorphism, [22, Corollary 1.2.6]. However, in this case the two norms coincide, as always, when  ·  is a Banach algebra norm on A0 (see also Lemma 5.1.2). Originally, the proper CQ*-algebras were introduced as a subcase of a richer structure, where three different involutions were involved. Now we shall discuss the essential features of CQ*-algebras in their original formulation. Definition 5.2.2 Let A# be a C*-algebra, with norm  · # and involution #. Let A[ · ] be a left Banach module over the C*-algebra A# , with isometric involution ∗, such that A# ⊂ A. Set A = (A# )∗ . We say that {A, ∗, A# , #} is a CQ*-algebra if (i) (ii) (iii) (iv)

A# is dense in A with respect to its norm  · ; A0 ≡ A# ∩ A is dense in A# with respect to its norm  · # ; (xy)∗ = y ∗ x ∗ , ∀ x, y ∈ A0 ; x# = sup xa, x ∈ A# . a∈A,a≤1

5.2 General CQ*-Algebras

103

Since ∗ is isometric, it is easy to see that the space A itself is a C*-algebra with respect to the norm x := x ∗ # and the involution x  := ((x ∗ )# )∗ . Moreover, notice that for every CQ*-algebra {A, ∗, A# , #}, the pair (A[ · ], A0 ) is a Banach quasi *-algebra. This follows by the very definitions after some technical calculations. Remark 5.2.3 It is quite clear that we can restate the previous definition starting from a C*-algebra A and a right module A over A , with A ⊂ A, satisfying the following properties: (i ) (ii ) (iii ) (iv )

A is dense in A with respect to its norm  · ; A0 ≡ A ∩ A# is dense in A with respect to its norm  ·  ; (xy)∗ = y ∗ x ∗ , ∀ x, y ∈ A0 ; x = sup ax, x ∈ A . a∈A,a≤1

It is then also natural to adopt the notation {A, ∗, A , } for indicating a CQ*-algebra as it has been done in many papers on this subject. Remark 5.2.4 Let {A, ∗, A# , #} be a CQ*-algebra. If A# has a unit e# , then e := e#∗ is a unit for A . We say that {A, ∗, A# , #} has a unit if the quasi *-algebra (A, A0 ) has a unit e. In this case, e is a unit for both A# and A . The situation for the structure of a CQ*-algebra is illustrated in the diagram of Fig. 5.1, where each arrow denotes a continuous embedding. According to Definition 5.1.1, a proper CQ*-algebra is then a CQ*-algebra, with A# = A = A0 and ∗ = # on A0 . There are several situations where a scheme like that of Fig. 5.1 below, can be constructed. However, the density conditions required in Definition 5.2.2 are not always fulfilled. We illustrate this point with the two following examples. Example 5.2.5 (Operators on Scales of Hilbert Spaces) Let H be a Hilbert space with scalar product ·|· and λ(., .) a positive sesquilinear closed form defined on a dense domain Dλ ⊂ H. Then, Dλ becomes a Hilbert space, that we denote by H+1 , Fig. 5.1 Structure of a CQ*-algebra

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with respect to the scalar product f |g+1 = f |g + λ(f, g),

∀ f, g ∈ Dλ .

(5.2.3)

The representation theorem for sesquilinear forms implies [16, Ch. VI, Sect. 2] the existence of a positive selfadjoint operator H such that D((I + H )1/2 ) = Dλ = H+1 ⊆ H and f |g+1 = (I + H )1/2 f |(I + H )1/2 g,

∀ f, g ∈ Dλ .

(5.2.4)

Let H−1 be the Hilbert space of conjugate linear functionals on H+1 . Its natural norm is denoted by ·−1 . The operator S = (1+H )1/2 has a bounded inverse S −1 , which maps H into Hλ . As a result, we can write: f |g+1 = Sf |Sg+1 = Uf |Ug−1

∀ f, g ∈ H+1

Here U is the operator from H+1 onto H−1 , whose existence is ensured by the Riesz lemma. This is the canonical way to get a scale of Hilbert spaces [20, VIII.6] i

j

H+1 → H → H−1 ,

(5.2.5)

where H−1 is the conjugate dual space of H+1 and i, j are continuous embeddings with dense range. In fact, the identity map i embeds H+1 in H and the map j : ψ ∈ H → j (ψ) ∈ H−1 , where j (ψ)(φ) := φ|ψ for all φ ∈ H+1 , is a linear embedding of H into H−1 . Identifying H+1 and H with their respective images in H−1 we can read (5.2.5) as a chain of topological inclusions H+1 ⊂ H ⊂ H−1 Let B(H+1 , H−1 ) be the Banach space of bounded operators from H+1 into H−1 and let us denote with T +1,−1 the natural norm of T in B(H+1 , H−1 ). We can introduce an involution in B(H+1 , H−1 ), as follows: to each element T ∈ B(H+1 , H−1 ) we associate the linear map T ∗ from H+1 into H−1 defined by the equation T ∗ f |g = T g|f ,

∀ f, g ∈ H+1

As it can be easily proved T ∗ ∈ B(H+1 , H−1 ) and T ∗ +1,−1 = T +1,−1 , for every T ∈ B(H+1 , H−1 ). Let B(H+1 ) denote the C ∗ -algebra of bounded linear operators on H+1 (the usual involution of B(H+1 ) will be denoted here by  ) and B(H−1 ) the C ∗ -algebra of bounded operators on H−1 (the natural involution of B(H−1 ) is denoted by #). Then, both B(H+1 ) and B(H−1 ) are contained in B(H+1 , H−1 ) and T ∈ B(H+1 ), if and

5.2 General CQ*-Algebras

105

only if, T ∗ ∈ B(H−1 ). Moreover (S  )∗ = (S ∗ )# , for every S ∈ B(H+1 ). The following subspace of B(H+1 , H−1 )   B + (H+1 ) := T ∈ B(H+1 , H−1 ) : T , T ∗ ∈ B(H+1 ) is a *-algebra, as it is easily seen. Then, one can show that B(H+1 , H−1 ) is a Banach module over the normed algebra B + (H+1 ), that can be viewed as a locally convex (Banach) partial *-algebra. This family of spaces realizes the situation depicted in Fig. 5.1, when one takes • • • •

A = B(H+1 , H−1 ), a Banach space; A = B(H+1 ), a C*-algebra; A# = B(H−1 ), also a C*-algebra; A0 = B + (H+1 ).

However, as proved in [2] it is not a CQ*-algebra, because the density conditions (i) and (ii) of Definition 5.2.2 are never satisfied. It is only possible to prove the existence of a maximal CQ*-algebra contained in it. This is obtained by completing B + (H+1 ) with respect to the norm  · +1,−1 and using Proposition 5.3.1 of the next subsection. Example 5.2.6 (Left Hilbert Algebras) For reader’s convenience, we briefly review here the definition and the basic properties of left Hilbert algebras. A *-algebra A0 with involution # is said to be a left Hilbert algebra [25, Section 10.1] if it is a dense subspace in a Hilbert space H with inner product ·|· satisfying the following conditions: (i) (ii) (iii) (iv)

for any x ∈ A0 , the map y ∈ A0 → xy ∈ A0 is continuous; xy|z= y|x # z, ∀x, y, z ∈ A0 ; A20 ≡ xy : x, y ∈ A0 is total in H; The involution x → x # is closable in H.

By (i), for any x ∈ A0 , denote by Lx , the unique continuous linear extension to H of the map y ∈ A0 → xy ∈ A0 ; then, using (ii), it is easy to see that the map L : x ∈ A0 → Lx ∈ B(H) is a bounded *-representation of A0 on H. Denote by S the closure of the operator S0 defined on the dense domain A20 by y ∈ A20 → y # ∈ H. 1

Let S = J  2 be the polar decomposition of S. Then, J is an isometric involution on H and  is a non-singular positive selfadjoint operator in H, such that S = 1 1 1 1 J  2 = − 2 J and S ∗ = J − 2 =  2 J ; J is called the modular conjugation operator of A0 and  is called the modular operator of A0 .

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5 CQ*-Algebras

We introduce now the commutant A0 of A0 as follows: for any y ∈ D(S ∗ ) we define a linear map Ry : D(S ∗ ) → H by Ry x := Lx y, x ∈ A0 , and we put   A0 := y ∈ D(S ∗ ) : Ry is bounded . Then, A0 is a right Hilbert algebra in H with involution y → y  := S ∗ y and multiplication y1 y2 ≡ Ry2 y1 , y1 , y2 ∈ A0 .  We want to warn the reader that we use here the word commutant (resp.,

bicommutant) and the symbol  (resp.,  ) to call and denote both the (von Neumann) commutant of a family of bounded operators (see Example A.1.7) and the “commutant” of a left Hilbert algebra as defined above: we follow, in fact, an established tradition.

We recall that a right Hilbert algebra B0 is defined similarly to the left one: B0 is a *-algebra with involution , which is a dense subspace in a Hilbert space H, with inner product ·|· satisfying the following conditions: (i ) (ii ) (iii ) (iv )

for any y ∈ B0 , the map x ∈ B0 → xy ∈ B0 is continuous; xy|z = x|zy  , ∀ x, y, z ∈ B0 ; B20 ≡ xy : x, y ∈ A is total in H; the involution y → y  is closable in H.

Now, the bicommutant A0 of A0 is defined by   A0 = x ∈ D(S) : y ∈ A0 → xy ∈ H is continuous . For any x ∈ A0 we denote by Lx the unique continuous linear operator on H, such that Lx y := Ry x, y ∈ A0 . Then, A0 is a left Hilbert algebra in H with involution S and multiplication x1 x2 ≡ Lx1 x2 , containing A0 . A left Hilbert algebra A0 is said to be full or achieved if A0 = A0 . It is well-known, as the Tomita fundamental theorem says, that J L(A0 )J = L(A0 ) and it L(A0 ) −it = L(A0 ) , for every t ∈ R; here L(A0 ) = {Lx , x ∈ A0 }, with L as in Example 5.2.6. Let A0 be a full left Hilbert algebra in H, and    A00 ≡ x ∈ D(α ) : α x ∈ A0 , ∀ α ∈ C . α∈C  in H, Then, A00 is a left Hilbert subalgebra  α  such that A00 = A0 , J A00 = A00 [27, VI, p. 22, Theorem 2.2];  : α ∈ C is a complex one-parameter group of automorphisms of A00 , such that

(α x)# = −α x # and (α x)∗ = −α x ∗ ,

∀ α ∈ C and x ∈ A00 .

The left Hilbert subalgebra A00 is called the maximal Tomita algebra of A0 . For more details, we refer to [25–27, 90].

5.2 General CQ*-Algebras

107

Let now A0 be a full left Hilbert algebra with unit e and involution # in H. Then, as seen above, the commutant A0 of A0 is a full right Hilbert algebra in H with (the same) unit and involution . The involution in H is defined by the modular conjugation operator J . For short we put H = A0 and H# = A0 . Consider now the system {H, J, H , } and introduce a topological structure in it. We start by defining for y ∈ H , y ≡ Ry  = sup xy, x≤1

where R denotes the regular *-representation of A0 in B(H). We also define x# := J x , for every x ∈ H# . Is {H, J, H# , #} a CQ*-algebra? First of all, we observe that conditions (i) and (iv) of Definition 5.2.2 are obviously fulfilled, whereas condition (iii) follows from the known equality (J x) = J x # , for every x ∈ H# . As for (ii), the C*-property for the norm  ·  is easily obtained from the fact that the linear map y → Ry is a *-representation of H into B(H). To show the completeness of H = A0 , one has to take into account the equality:   A0 = y ∈ D(S ∗ ) : Ry is bounded . Now, if {yk } is a  ·  -Cauchy sequence in H , since e ∈ A0 , one can find an element y ∈ H, such that yk converges to y with respect to the Hilbert norm; moreover, since as is known, for each y ∈ A0 , y  = S ∗ y, the sequence {S ∗ yk } is also convergent. Therefore, y ∈ D(S ∗ ). The fact that Ry is bounded follows easily from the norm completeness of B(H). To conclude that {H, J, H# , #} is a CQ*-algebra, we should prove the density of H ∩ H# in H with respect to  ·  . This question is still open; however in [25, Section 10.19] it is shown that the set 

 fr ()fs (−1 )y : y ∈ H , r, s > 0 ,

where fm (x) = exp (−mx) and  is the modular operator, is contained in H ∩ H# . This set is, in a sense, quite rich; indeed, a simple application of the spectral theorem for the operator  and the Lebesgue dominated convergence theorem shows that fr ()fs (−1 )y converges to y with respect to the Hilbert norm, for each y ∈ H . A deeper analysis of these points can be found in Sect. 5.5. Remark 5.2.7 In Chap. 3, given a Banach quasi *-algebra (A[ · ], A0 ), an important role has been played by the family of sesquilinear forms SA0 (A) of Definition 3.1.14. As we have noticed, from the very definitions follows that, if {A, ∗, A# , #} is a CQ*-algebra, then (A[·], A0 ) is a Banach quasi *-algebra. Hence, it makes sense to consider the set SA0 (A) also in this case. The density properties required in Definition 5.2.2 make stronger the condition (ii) of Definition 3.1.14. In

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fact, the following condition holds: for every ϕ ∈ SA0 (A), ϕ(xa, y) = ϕ(x, ya ∗ ),

∀ a ∈ A, x, y ∈ A# ;

(5.2.6)

ϕ(ax, y) = ϕ(x, a ∗ y),

∀ a ∈ A, x, y ∈ A .

(5.2.7)

For this reason, when dealing with a CQ*-algebra {A, ∗, A# , #} we shall not distinguish with a different notation the family of sesquilinear forms satisfying (5.2.6) or (5.2.7), which are equivalent.

5.3 Construction of CQ*-Algebras 5.3.1 Constructive Method The next proposition extends the constructive Proposition 5.1.3. Proposition 5.3.1 Let A# be a C*-algebra, with norm  · # and involution #; let  ·  be another norm on A# , weaker than  · # and such that (i) xy ≤ x# y, ∀ x, y ∈ A# ; (ii) there exists a  · # -dense subalgebra A0 of A# , where an involution * (which makes A0 into a *-algebra) is defined with the property x ∗  = x,

∀ x ∈ A0 .

Let A be the  · -completion of A# . Then, {A, ∗, A# , #} is a CQ*-algebra. Proof Since A0 is  · # -dense in A# , then the  · -completions of A0 and A# can be identified with the same Banach space A. Now, for x ∈ A# , put x# = sup xa. a≤1

(5.3.8)

Because of (i) we have x# ≤ x# ,

∀ x ∈ A# .

The converse inequality follows from Lemma 5.1.2.

 

Corollary 5.3.2 Let A# be a C*-algebra and A a left Banach A# -module with involution ∗ and B0 any *-subalgebra of A∗# ∩ A# , which is also #-invariant. Let M(B0 )# be the closure of B0 in A# and M(B0 ) the closure of B0 in A. Then, {M(B0 ), ∗, M(B0 )# , #} is a CQ*-algebra.

5.3 Construction of CQ*-Algebras

109

Proof We notice that M(B0 )# is a C*-algebra, with respect to the involution # and the norm ·# , since B0 is an involutive algebra also with respect to #. The statement then follows from the previous proposition.   Let us now give some explicit applications of Proposition 5.3.1. Example 5.3.3 Let S be an unbounded selfadjoint operator in a Hilbert space H with domain D(S) and with bounded inverse S −1 in B(H), such that S −1  ≤ 1. We define the commutant of the operator S −1 , as follows  (S −1 ) := T ∈ B(H) : T S −1 = S −1 T .

(5.3.9)

It is straightforward to check that (S −1 ) is a C*-algebra (indeed, a von Neumann algebra), being a norm closed *-subalgebra of B(H). Moreover, if S has not simple spectrum, (S −1 ) is not abelian [1, II, Ch.VI, n.69]. Define a norm  · C on (S −1 ) , weaker than the operator norm on B(H), in the following way T C := S −1 T S −1  = S −2 T  = T S −2 .

(5.3.10)

If we call C the  · C -completion of (S −1 ) , Proposition 5.3.1 ensures us that (C[ · C ], (S −1 ) ) is a proper nonabelian CQ*-algebra. The non triviality of the construction follows from the fact that  · C is not equivalent to  · , as it is easily checked. We now prove the *-semisimplicity of (C[ · C ], (S −1 ) ). First we observe that any sesquilinear form of the following type ϕξ (A, B) = S −1 AS −1 ξ |S −1 BS −1 ξ , A, B ∈ C[ · C ], belongs to S(S −1 ) (C), for any ξ ∈ H, with ξ  ≤ 1. Therefore, if ϕ(A, A) = 0, for all ϕ ∈ S(S −1 ) (C), it follows, in particular, that ϕξ (A, A) = 0, for every ξ ∈ H. This implies that S −1 AS −1 = 0 and therefore that A = 0. Example 5.3.4 As before, let H be a Hilbert space with scalar product ·|· and consider the triplet of Hilbert spaces H+1 ⊂ H ⊂ H−1 as generated by a positive selfadjoint operator S, with dense domain D(S) and S ≥ I , as in Example 5.2.5. In that example it was claimed that {B(H+1 , H−1 ), ∗, B(H−1 ), #} is never a CQ*algebra. Nevertheless, Proposition 5.3.1 provides a canonical way of constructing a CQ*-algebra of operators acting in the given scale of Hilbert spaces. Indeed, since B + (H+1 ) ⊂ B(H−1 ), we may consider the largest *-subalgebra B0 of B + (H−1 ), which is also invariant with respect to the involution # and define Bc (H+1 ) as the C*-subalgebra of B(H−1 ) generated by B0 . The non triviality of this set is discussed in [2, Section 10.4], where a characterization of B0 is also given. Then, the conditions of Proposition 5.3.1 are fulfilled, by choosing the weaker norm on Bc (H+1 ) as equal to  · +1,−1 . Therefore, if we denote with Bc (H+1 , H−1 ) the subspace of B(H+1 , H−1 ) obtained by completing Bc (H−1 ) with respect to the norm  · +1,−1 we obtain the CQ*-algebra {Bc (H+1 , H+1 ), ∗, Bc (H+1 ), #} .

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Example 5.3.5 (CQ*-Algebras of Compact Operators) The same approach can be repeated starting from any C*-subalgebra Q of Bc (H+1 ), since conditions (i) and (ii) of Proposition 5.3.1 are satisfied, whenever the weaker norm  ·  is just  · +1,−1 and the adjoint is the one in B(H+1 , H−1 ). In particular, we give now an example, with H separable, where all the spaces involved are explicitly identified. We start introducing the following sets of operators  A = T ∈ B(H+1 , H−1 ) : S −1 T S −1 is compact in H ,  A = T ∈ B(H+1 ) : ST S −1 is compact in H ,  A# = T ∈ B(H−1 ) : S −1 T S is compact in H . These sets are non empty: for instance, A contains any operator of the form SZS, with Z compact in H. As in the previous example, we indicate with the same symbol, S, both the operator from H+1 into H and its extension from H into H−1 . The sets above coincide with the following ones:   A = T ∈ B(H+1 , H−1 ) : T is compact from H+1 into H−1 ,   A = T ∈ B(H+1 ) : T is compact in H+1 ,   A# = T ∈ B(H−1 ) : T is compact in H−1 . It is easy to check that A is a C*-algebra with respect to the involution  and to the norm  ·  = S · S −1 . Analogously A# is a C*-algebra with respect to the involution # = ∗∗ and to the norm  · # = S −1 · S, while A is a Banach space with respect to the involution ∗ and to the norm · = S −1 ·S −1 . The norms · ,  · # and  ·  coincide with those defined in {B(H+1 , H−1 ), ∗, B(H−1 ), #} and the involutions # and ∗ are the ones defined respectively in B(H−1 ) and B(H+1 , H−1 ). In order to prove the density conditions, let us consider the family of projection operators Pξ , ξ ∈ H+1 with ξ  = 1 (the norm in H) defined by Pξ η = η|ξ ξ,

η ∈ H.

Each operator Pξ has an obvious extension to H−1 , which we still call Pξ . It is straightforward to prove that Pξ ∈ B(H+1 ), Pξ ∈ B(H−1 ) and Pξ ∈ A . Let A0 be the subalgebra of A generated by all the operators Pξ , ξ ∈ H+1 . This is closed with respect to the adjoint ∗ and it is also  ·  -dense in A , since any compact operator is the norm limit of operators of finite rank. Moreover, it is also  · -dense in A. Applying Proposition 5.3.1, we conclude that {A, ∗, A , } is a CQ*-algebra of operators.

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5.3.2 Starting from a *-Algebra Let A0 [ · ] be a *-algebra endowed with a norm  · . Suppose that the involution ∗ is isometric with respect to  ·  and the multiplication is separately (but not jointly) continuous. Then, as we have seen in the previous sections, the pair (A[ · ], A0 ), 0 [ · ] of A0 [ · ], is a Banach quasi *-algebra. where A[ · ] is the completion A As before, for any a ∈ A, we put La x = ax and Ra x = xa, x ∈ A0 . Then, La and Ra are linear maps of A0 into A[ · ]. In particular, if z ∈ A0 , then Lz and Rz can be extended to bounded linear operators on the Banach space A[ · ] and they are denoted by the same symbols Lz and Rz . Let (A[ · ], A0 ) be a Banach quasi *-algebra and assume that the *-algebra A0 has another norm  · # and another involution # satisfying the following conditions: (a.1) x # x# = x2# , ∀ x ∈ A0 ; (a.2) x ≤ x# , ∀ x ∈ A0 ; (a.3) xy ≤ x# y, ∀ x, y ∈ A0 . Then, by (a.2), the identity map i : A0 [ · # ] → A0 [ · ] has a continuous extension  i from the completion A# of A[ · # ] (A# is, of course, a C*-algebra) into A[ · ]. If  i is injective, then A# is (identified with) a dense subspace of A. This happens, if and only if, (a.4) the two norms  ·  and  · # are compatible in the following sense [11]: for any sequence {xn } ⊂ A0 , such that xn  → 0 and xn → x in A# [ · # ], one obtains x = 0; i.e., this happens when  i −1 : A0 [ · ] → A# [ · # ] is closable. Definition 5.3.6 A Banach quasi *-algebra (A[ · ], A0 ) is said to be a pseudo CQ*-algebra, if the *-algebra A0 has another norm  · # and another involution # satisfying the conditions (a.1)–(a.4) above. A pseudo CQ*-algebra (A[ · ], A0 ) is said to be a strict CQ*-algebra, if x# = Lx  ≡ sup{xy : y ∈ A0 with y ≤ 1},

∀ x ∈ A0 .

(5.3.11)

Proposition 5.3.7 Let (A[·], A0 ) be a strict CQ*-algebra and A# the C*-algebra obtained by completing A0 with respect to  · # . Then, {A, ∗, A# , #} is a CQ*algebra. Conversely, if {A, ∗, A# , #} is a CQ*-algebra and A0 := A# ∩ A , then (A[ · ], A0 ) is a strict CQ*-algebra. Proof As discussed before, if (A[ · ], A0 ) is a pseudo CQ*-algebra, then the completion A# of A0 [ · # ] is a C*-algebra identified with a subspace of A (which we call A# too) and the same holds, of course, for A := A∗# , with respect to the norm x := x ∗ # . The density properties required in Definition 5.2.2 can be derived from the construction itself. It remains to show that (iv) of Definition 5.2.2 holds,

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that is x# =

sup

xa,

∀ x ∈ A# .

(5.3.12)

a∈A,a≤1

But this follows easily from the density properties of A0 and from (5.3.11). Conversely, if {A, ∗, A# , #} is a CQ*-algebra and A0 := A# ∩ A , then the pair (A[ · ], A0 ) is a Banach quasi *-algebra. It is clear that the norm  · # and the involution # of A# , restricted to A0 , satisfy the conditions (a.1)–(a.3) above. As for (a.4), we just notice that, for any sequence {xn } ⊂ A0 , such that xn  → 0 and xn − x# → 0, for some x ∈ A# , we necessarily have x = 0, since y ≤ y# , for every y ∈ A# .    A pseudo CQ*-algebra (A[ · ], A0 ) is fully determined by the involution # and

the C*-norm ·# . For this reason, it will be often denoted by (A[·], #, A0 , ·# ), in the case when it will be necessary to indicate explicitly the different norms.

 A strict CQ*-algebra is fully determined, when the new involution # is known;

so it can be simply denoted by (A[ · ], A0 , #).

Let (A[ · ], #, A0 ,  · # ) be a pseudo CQ*-algebra and, as above, A# the C*algebra obtained by completing the #-algebra A0 with respect to the C*-norm  · # . Let JA be the involution ∗ of the Banach quasi *-algebra (A[ · ], A0 ). Then, as noted before, A ≡ JA A# is a C*-algebra with involution x  ≡ x ∗#∗ where x ∗#∗ = ∗ # ∗ ∗ and C*-norm x (x )   ≡ x# ,for all x ∈ A .   If M ⊂ A# let LM = Lx : x ∈ M and if N ⊂ A let RN = Ry : y ∈ N . Proposition 5.3.8 A pseudo CQ*-algebra (A[ · ], #, A0 ,  · # ) contains two C*algebras A# and A ≡ JA A# with different involutions # and , respectively, as dense subalgebras. In particular, if (A[ · ], A0 , #) is a strict CQ*-algebra, then LA# and RA are C*-algebras, Lx Ry = Ry Lx , for each x ∈ A# , y ∈ A and RA = JA LA# JA . We summarize the situation with the following scheme ⊂

A# ⊂ "J A[ · ] ⊂ ⊂ A *-algebra, with norm C*-algebras pseudo CQ*-algebra, A0 [ · ]

0 [ ·  ] and A# = i.e., the *-algebra A0 [ · ] is contained in its closure A = A  A0 [ · # ] = J A . Moreover, these C*-algebras are both contained in A[ · ]. We shall come back to pseudo CQ*-algebras and strict CQ*-algebras in Sect. 5.5.

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5.3.3 Construction of CQ*-Algebras through Families of Forms In Sect. 3.1.3 we discussed *-semisimple Banach quasi *-algebras and we obtained several consequences of this definition; for *-semisimplicity of CQ*-algebras, see discussion before Proposition 5.4.7. Here we introduce a different definition of semisimplicity, the #-semisimplicity, that allows to define a new norm satisfying the assumptions of Proposition 5.3.1. The application of this proposition leads to the construction of a new CQ*-algebra, whose norm closely reminds the characterization of a C*-norm in terms of states given by Gelfand. Definition 5.3.9 Let {A, ∗, A# , #} be a CQ*-algebra. We denote as T# (A) the set of sesquilinear forms ϕ on A × A with the following properties: (i) (ii) (iii) (iv)

ϕ(a, a) ≥ 0, ∀ a ∈ A; ϕ(xa, b) = ϕ(a, x # b), ∀ a, b ∈ A, ∀ x ∈ A# ; |ϕ(a, b)| ≤ a b, ∀ a, b ∈ A; ϕ(a, b) = ϕ(b∗ , a ∗ ), ∀ a, b ∈ A.

The CQ*-algebra {A, ∗, A# , #} is called #-semisimple, if a ∈ A with ϕ(a, a) = 0, for all ϕ ∈ T# (A), implies a = 0. It is worth remarking that while conditions (i) and (iii) were already present in the definition of the family SA0 (A), condition (iv) is peculiar for this family of forms, and (ii) is a natural modification of that for SA0 (A). The non triviality of this definition will appear clearer in Sect. 5.5, where it is shown how the Tomita– Takesaki theory naturally provides examples of sesquilinear forms of this kind. In a very similar way, we could speak of -semisimplicity, simply starting with a family of sesquilinear forms T (A), where (ii) is replaced by the specular condition (ii ) ϕ(ay, b) = ϕ(a, by  ),

∀ a, b ∈ A and y ∈ A ,

while the other conditions are kept fixed. However, due to condition (iv) and to the equality (x  )∗ = (x ∗ )# , for each x ∈ A , it is easily seen that the two families T (A) and T# (A) coincide. By the way, it is also interesting to remark that without condition (iv) in the definition of the two preceding sets, their equality is replaced by a weaker, but still interesting, result: there is a one-to-one correspondence between their sesquilinear forms, given by T (A) # ϕ → ϕ ∗ ∈ T# (A) with ϕ ∗ (a, b) ≡ ϕ(b, a),

∀ a, b ∈ A.

It is easy to check that ϕ ∈ T (A), if and only if, ϕ ∗ ∈ T# (A). It is evident that, either if condition (iv) is assumed or not, -semisimplicity and #-semisimplicity are equivalent. One of the main reasons for the introduction of the set T# (A) is that it allows, by means of the constructive Proposition 5.3.1, to build up examples of CQ*-algebras

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starting with a given #-semisimple CQ*-algebra {A, ∗, A# , #}. First, we introduce on A a new norm aT :=

sup ϕ(a, a)1/2 , a ∈ A.

ϕ∈T# (A)

(5.3.13)

It is not difficult to see that this is really a norm. In particular, the #-semisimplicity implies that aT = 0, if and only if, a = 0. Property (iv) of the family T# (A) implies that aT = a ∗ T , a ∈ A. Furthermore, condition (iii) implies that aT ≤ a, for any a ∈ A. In order to apply Proposition 5.3.1 we still have to check that the following inequality holds xyT ≤ x# yT ,

∀ x, y ∈ A# .

(5.3.14)

Indeed, defining ω(x) := ϕ(x, e), for x ∈ A# , we get ω(x # x) ≥ 0. This implies that ω is also continuous and that the following inequality holds, |ω(y # xy)| ≤ ω(y # y)x# ,

∀ x, y ∈ A# .

Now, inequality (5.3.14) is an immediate consequence of the definition of  · T . Applying Proposition 5.3.1 we can conclude that {AT , ∗, A# , #}, where AT is the completion of A# under the norm  · T , is a CQ*-algebra, containing {A, ∗, A# , #}. Indeed, A ⊂ AT , since both A and AT are completions of the same C*-algebra A# with respect to the norms  ·  and  · T that satisfy the condition  · T ≤  ·  and are compatible in the sense of [11]; this means, that it is possible to extend by continuity the identity map i : A [ · ] → A [ · T ] to the respective completions of the normed algebras involved: i.e.,  i : A[ · ] → AT [ · T ], preserving its injectivity. Let us now prove the following Lemma 5.3.10 Let {A, ∗, A# , #} be a #-semisimple CQ*-algebra. Then, T# (A) = T# (AT ). Proof Let ϕ ∈ T# (AT ). We call ϕr the restriction of ϕ to A × A. Since  · T is weaker than  · , then ϕr belongs to T# (A). Conversely, if ϕ ∈ T# (A), then because of the following bound, |ϕ(a, b)| ≤ ϕ(a, a)1/2 ϕ(b, b)1/2 ≤ aT bT , for all a, b ∈ A, ϕ can be extended to AT × AT and still satisfies all the required properties.   It is evident from the proof that the equality of the two sets must be understood as the possibility of associating a form of T# (AT ) to a form of T# (A) and vice versa. Proposition 5.3.11 The CQ*-algebra {AT , ∗, A# , #} is #-semisimple.

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Proof Let a ∈ A with ϕ(a, a) = 0, for every ϕ ∈ T# (AT ). Then, due to the above Lemma and to the #-semisimplicity of {A, ∗, A# , #}, we conclude that a = 0.   With the previous construction, one can always construct, starting from a #semisimple CQ*-algebra {A, ∗, A# , #}, a new CQ*-algebra, whose norm has the form (5.3.13), a property that closely reminds what happens for C*-algebras. We now give another similar construction, starting this time from a family of sesquilinear forms on the C*-algebra A# . Let {A, ∗, A# , #} be a CQ*-algebra with unit e. We denote with # the family of all sesquilinear forms ϕ of the C*-algebra A# satisfying the following properties: (c.1) (c.2) (c.3) (c.4)

ϕ(x, x) ≥ 0, ∀ x ∈ A# ; ϕ(xy, z) = ϕ(y, x # z), ∀ x, y, z ∈ A# ; ϕ(e, e) ≤ 1; ϕ(x, y) = ϕ(y ∗ , x ∗ ), ∀ x, y ∈ A# ∩ A .

Remark 5.3.12 If {A, ∗, A# , #} is a CQ*-algebra with unit e, then there is a one-toone correspondence between the sesquilinear forms ϕ of A# satisfying the conditions (c.1)–(c.3) (denote the respective family with F) and the positive linear functionals ω on A# fulfilling the property ω ≤ 1. Then, if we define xF := sup ϕ(x, x)1/2 , ϕ∈F

x ∈ A# ,

which is analogous to x below, we get exactly  · # (in this respect, see also Theorem A.5.5). Considering the adjoint forms ϕ ∗ , we get a result for A , similar to the preceding one. So what makes the difference is condition (c.4). Furthermore, assume that {A, ∗, A# , #} satisfies the following property: If x ∈ A# and ϕ(x, x) = 0,

∀ ϕ ∈ # , then x = 0.

Since for each x ∈ A# the linear functional ωy (x) = ϕ(xy, y),

y ∈ A# ,

is positive, one can easily prove the inequality |ϕ(xy, y)| ≤ ϕ(y, y)x# ,

∀ x, y ∈ A# ,

and then that |ϕ(x, y)| = |ϕ(y # x, e)| ≤ x# y# ,

∀ x, y ∈ A# .

(5.3.15)

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We now can define a new norm on A# by x := sup ϕ(x, x)1/2 ,

x ∈ A# .

ϕ∈#

This norm defines on A# a topology coarser than that of the original norm. With the help of the previous inequalities and of (c.4), we can prove that xy ≤ x# y ,

∀ x, y ∈ A# ,

and x ∗  = x ,

∀ x ∈ A# ∩ A .

Thus Proposition 5.3.1 applies and if  A denotes the completion of A# [ ·  ], then { A, ∗, A# , #} is a CQ*-algebra. Now, it is easy to see that if ϕ ∈ T# ( A), then ϕA# is an element of # . Conversely, if ϕ0 ∈ # , since ϕ0 (x, y) ≤ x y ,

∀ x, y ∈ A# ,

A denoted by ϕ. It is easy to check that then ϕ0 has a continuous extension to  ϕ ∈ T# ( A). However, in spite of this very close relation between the sets # and T# ( A), { A, ∗, A# , #} need not be #-semisimple. Remark 5.3.13 Every C*-algebra with unit possesses sesquilinear forms satisfying the properties (c.1)–(c.3), but not necessarily the property (c.4); there are in fact C*algebras, which do not have any nonzero trace. On the other hand, if (c.1)–(c.4) and condition (5.3.15) hold, the previous construction shows that it is possible to define a new CQ*-algebra, where each element of # is bounded.

5.4 *-Homomorphisms of CQ*-Algebras The automatic continuity of *-homomorphisms and the uniqueness of the C*-norm are certainly among the most important features of the theory of C*-algebras (see, e.g., Sect. A.6.2). These properties, however, are not preserved in the framework of CQ*-algebras. In particular, as we shall see below, there exist non equivalent norms on a quasi *-algebra (A, A0 ), with A0 a C*-algebra, that make it into topologically different CQ*-algebras. To address this question, we begin with introducing a convenient notion of a *-homomorphism for CQ*-algebras [51].

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117

Definition 5.4.1 Let {A, ∗, A# , #}, {B, ∗, B# , #} be two CQ*-algebras. A linear map

: A → B is said to be a *-homomorphism of {A, ∗, A# , #} into {B, ∗, B# , #} if (i) (a ∗ ) = (a)∗ , ∀ a ∈ A; (ii) # :=  A# maps A# into B# and is a *-homomorphism of C*-algebras; (iii) (xa) = (x) (a), ∀ a ∈ A, x ∈ A# . The *-homomorphism # from A# into B# defines a *-homomorphism  from the C*-algebra A into the C*-algebra B by  (y) := # (y ∗ )∗ , y ∈ A . Clearly  is a *-isomorphism of C*-algebras, if and only if, # is a *-isomorphism.  Observe that, in what follows, for simplicity’s sake, we use the same symbol  · 

for the norms of both Banach spaces A and B.

A bijective *-homomorphism , such that (A# ) = B# , is called a *isomorphism. Continuity of *-homomorphisms is, of course, equivalent to their boundedness. We will adopt the following terminology. A *-homomorphism is called contractive if  (a) ≤ a, for every a ∈ A. A *-homomorphism can be continuous without being contractive. A contractive *-isomorphism, whose inverse is also contractive is called an isometric *-isomorphism. If is a *-homomorphism, then  # (x)# ≤ x# , for every x ∈ A# , since each *-homomorphism of C*-algebras is contractive. Analogously, if # is a *isomorphism, then it is necessarily isometric (Proposition A.6.13). Finally, we notice that, if is a *-isomorphism and {A, ∗, A# , #}, {B, ∗, B# , #} have units eA , eB , respectively (see Remark 5.2.4), then by (iii), (eA ) = eB . Taking into account that a continuous invertible linear map from the Banach space A onto the Banach space B has a continuous inverse, we get Proposition 5.4.2 Let be a continuous *-isomorphism from the CQ*-algebra {A, ∗, A# , #} onto the CQ*-algebra {B, ∗, B# , #}. Then, there exist δ, γ > 0, such that δa ≤  (a) ≤ γ a,

∀ a ∈ A.

On the other hand, we have the following Proposition 5.4.3 Let {A, ∗, A# , #} and {B, ∗, B# , #} be CQ*-algebras. Let # be a *-isomorphism of A# onto B# , for which there exist 0 < δ, γ , such that δx ≤  # (x) ≤ γ x, ∀ x ∈ A# .

(5.4.16)

Then, # can be continuously extended to a continuous *-isomorphism of A onto B, which is contractive, if γ ≤ 1. Moreover, if  # (x) = x, for all x ∈ A# , the extension is isometric.

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Proof For a ∈ A there exists a sequence {xn } ⊂ A# converging to a. Then, one defines

(a) :=  ·  − lim # (xn ). n→∞

It is easily seen that (a) is well-defined and satisfies the conditions of Definition 5.4.1. By (5.4.16) it follows that is a bijection and so it is a continuous *-isomorphism. By the same relation # is  · -homeomorphism, so the same also   holds for . Proposition 5.4.4 Let be a *-isomorphism from the CQ*-algebra (A, ∗, A# , #) onto the CQ*-algebra {B, ∗, B# , #}. Then, it is possible to define a new norm  ·  on B, such that {B, ∗, B# , #} is still a CQ*-algebra and is isometric. Proof We define b :=  −1 (b), for every b ∈ B. It is very easy to prove that (i) B[ ·  ] is a Banach space; (ii) b∗  = b , ∀ b ∈ B; (iii) B# is  ·  -dense in B. Let us now define a new norm on B# : sup xb , x ∈ B# , b ∈ B. x # := b ≤1

−1 is necessarily an isometry We prove that x# = x # , for all x ∈ B# . Since between A# and B# , for b ∈ B and x ∈ B# we have

xb =  −1 (xb) ≤  −1 (x)#  −1 (b) ≤ x# b . Therefore, x # ≤ x# , for all x ∈ B# . This inequality and the inverse mapping theorem imply that the two norms are equivalent. To show that they are exactly equal we can proceed as in the proof of Proposition 5.3.1, after checking that  ·  # makes of B# a normed algebra; i.e., we need first the inequality

xy # ≤ x# y# ,

∀ x, y ∈ B# .

But this can be easily derived from the definition of  ·  # itself. The equality

, for all x ∈ B , is easy to be proved. x #  = x   # # # From the previous proof one can also deduce the following

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Proposition 5.4.5 Let {A, ∗, A# , #} be a CQ*-algebra and B a left B# -module, where B# is a C*-algebra with involution # and norm  · # . Moreover, let B# ⊂ B and be an injective linear map from A into B with the properties: (i) (a ∗ ) = (a)∗ , ∀ a ∈ A; (ii) # :=  A# maps A# into B# and is a *-homomorphism of C*-algebras; (iii) (xa) = (x) (a), ∀ a ∈ A, x ∈ A# . Let us define a norm  ·  on (A) by b :=  −1 (b),

b ∈ (A).

Then, { (A)[ ·  ], ∗, B# , #} is a CQ*-algebra. Now we can answer the following question: given the CQ*-algebra {A, ∗, A# , #}, is it possible to endow A with a non equivalent norm, which makes of it a CQ*algebra on the same C*-algebra A# ? The following explicit construction shows that the answer is positive. In particular, we shall give a general strategy to build up different proper CQ*-algebras over the same C*-algebra and, after that, we shall give an explicit example. Our starting point is a proper CQ*-algebra (A[ · ], A0 ). The C*-norm of A0 is denoted again by  · 0 . Proposition 5.4.6 Let T be an unbounded, invertible linear map from A onto A, such that T A0 = A0 , T (x ∗ ) = (T x)∗ and T (xy) ≤ T xy0 , for all x, y in A0 . Then, defining a := T a, a ∈ A, (A[ ·  ], A0 ) is a proper CQ*-algebra with norm  ·  non equivalent to  · . Proof We begin with proving that A[ ·  ] is a Banach space. Let {an } be a  ·  Cauchy sequence in A. This implies that {T an } is a  · -Cauchy sequence in A, therefore it converges to an element b ∈ A. Set a = T −1 b. Then, an − a = T an − T T −1 b = T an − b → 0, which proves the statement. Now we show that A0 is dense in A[ ·  ]. Indeed, let a ∈ A and put b = T −1 a. Let {xn } be a sequence of elements of A0 , such that xn − a → 0. Then, we have T −1 xn − b = T T −1 xn − T b = T T −1 xn − T T −1 a = xn − a → 0. By the assumption it follows that {T −1 xn } ⊂ A0 . Hence A0 is dense in A[ ·  ]. The assumptions on T allow to construct a proper CQ*-algebra with the help of Proposition 5.1.3 by completing A0 with respect to  ·  . The completeness of A[ ·  ] and the density of A0 in it, proved above, imply that the outcome of this construction is (A[ ·  ], A0 ), which therefore is a proper CQ*-algebra.  

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It is worth noticing that, in this construction, the two Banach spaces A[ · ] and A[ ·  ] coincide, while the two norms,  ·  and  ·  differ, for all those elements belonging to A, but not to A0 , and for this reason they are not equivalent. An explicit example of an operator T , as above, can be constructed by means of the Hamel basis of the Banach spaces A and A0 . Indeed, let eα {α∈J } be a Hamel basis for A, which contains a Hamel basis {eα }α∈I , for A0 . We can always choose eα , such that eα = eα∗ . In order to simplify things, we suppose that the set J is a subset of the positive reals with no upper bound. We define T through its action on the basis vectors eα , i.e., T eα := eα , if α ∈ I and T eα := αeα , if α ∈ J \I. With this definition, it is clear that T is unbounded and invertible. Moreover, since T is the identity map on A0 , it is also evident that T (x ∗ ) = (T x)∗ , for all x ∈ A0 , as well as that the inequality T (xy) ≤ T xy0 holds, for all x, y in A0 . However, it is also clear that the norm · is not equivalent to ·, as one can check considering the values of these norms on the basis vectors. In this way, we have constructed an operator T satisfying all the properties required in Proposition 5.4.6. In the rest of this section we discuss briefly the problem of *-semisimplicity of CQ*-algebras in relation with *-isomorphisms. Note that when we say that a CQ*-algebra {A, ∗, A# , #} is *-semisimple, we mean that the Banach quasi *-algebra (A[ · ], A0 ), with A0 ≡ A# ∩ A , is *-semisimple (for the notation and the fact that (A[ · ], A0 ) is a Banach quasi *-algebra, see Definition 5.2.2 and the discussion after it). To begin with, let {A, ∗, A# , #} and {B, ∗, B# , #} be CQ*-algebras and : A → B a *-homomorphism of {A, ∗, A# , #} into {B, ∗, B# , #}. If ψ ∈ SB0 (B) and is contractive, then the sesquilinear form ψ ◦ defined by (ψ ◦ )(a, b) = ψ( (a), (b)), a, b ∈ A is an element of SA0 (A). Suppose that {B, ∗, B# , #} is *-semisimple and that ϕ(a, a) = 0, for every ϕ ∈ SA0 (A). Then in particular, for every ψ ∈ SB0 (B), (ψ ◦ )(a, a) = 0. Hence (a) = 0. Thus, if is injective, a = 0 (see also Definition 3.1.17 and Corollary 3.1.24). If is bounded but not contractive, we can replace it with  := / , which is obviously contractive. We have then proved the following Proposition 5.4.7 Let {A, ∗, A# , #} and {B, ∗, B# , #} be CQ*-algebras and : A → B an injective *-homomorphism of {A, ∗, A# , #} into {B, ∗, B# , #}. If {B, ∗, B# , #} is *-semisimple, then {A, ∗, A# , #} is *-semisimple. In particular, if

is a *-isomorphism, {A, ∗, A# , #} is *-semisimple, if and only if, {B, ∗, B# , #} is *-semisimple. If is an isometric *-isomorphism then the following equalities are easily proved:   SA0 (A) = ψ ◦ : ψ ∈ SB0 (B) ,

  SB0 (B) = ϕ ◦ −1 : ϕ ∈ SA0 (A) .

Corollary 5.4.8 Let {A, ∗, A# , #} and {B, ∗, B# , #} be CQ*-algebras with B ⊂ A and B# ⊂ A# . If B is continuously embedded in A and A is *-semisimple, then B is also *-semisimple.

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Proof It follows immediately from Proposition 5.4.7.

 

Remark 5.4.9 Let {A, ∗, A# , #} and {B, ∗, B# , #} be CQ*-algebras with units eA , eB , respectively. We can also introduce a more general notion than that of *homomorphism. A *-bimorphism of {A, ∗, A# , #} into{B, ∗, B# , #} is a pair (π, π# ) of linear maps π : A → B and π# : A# → B# , such that (i) π# is a homomorphism of algebras with π# (x # ) = π# (x)# , x ∈ A# ; i.e., π# is a #-homomorpism. (ii) π(a ∗ ) = π(a)∗ , ∀ a ∈ A; (iii) π(xa) = π# (x)π(a), ∀ a ∈ A, x ∈ A# . In general, the restriction of π to A# is different from π# . For this reason, *-homomorphisms and *-bimorphisms are different objects. Of course any *homomorphism defines, in trivial way, a *-bimorphism, but the converse is not true, in general. If (π, π# ) is a *-bimorphism, then π(a) = π(eA )π# (a), for each a ∈ A# ; but, in general, π(eA ) is different from eB . Obviously, if π(eA ) = eB , then π is a *-homomorphism. If (π, π# ) is a *-bimorphism of {A, ∗, A# , #} into (B, ∗, B# , #), then π(ax) = π(a)π (x),

∀ a ∈ A, x ∈ A ,

where π (x) = π# (x ∗ )∗ . Moreover, π is a homomorphism of A into B preserving the involution  of A .

5.5 CQ*-Algebras and Left Hilbert Algebras In this section we shall consider again CQ*-algebras constructed from left Hilbert algebras and we shall examine their link with pseudo (strict) CQ*-algebras of Sect. 5.3.2. The starting point is a Hilbertian quasi *-algebra (A[·], A0 ), as defined in Remark 3.1.9. Definition 5.5.1 A Hilbertian quasi *-algebra (A[ · ], A0 ) is said to be an HCQ*algebra if A is endowed with another involution #, such that L∗x = Lx # and x ≤ Lx , for each x ∈ A0 . A Hilbertian quasi *-algebra will be simply denoted by (A[ · ], #). HCQ*-algebras are closely related to left Hilbert algebras (see Example 5.2.6 for notations, basic definitions and facts). Proposition 5.5.2 Suppose that (A[ · ], #) is an HCQ*-algebra with involution operator JA ; i.e., JA a = a ∗ , a ∈ A. Then, the following statements hold: (i) (A[ · ], #) is a strict CQ*-algebra under the C*-norm x# = Lx , x ∈ A0 ; (ii) A0 is a left Hilbert algebra in the Hilbert space H ≡ A[ · ], whose full left Hilbert algebra A0 has a unit u.

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Proof (i) The proof is mostly trivial. We only prove that, in this case, condition (a.4) is satisfied, (see discussion before Definition 5.3.6). Indeed, if {xn } ⊂ A0 is a sequence such that xn  → 0 and xn → x in A# [ · # ], then Lxn → Lx , with respect to the operator norm. Continuity of multiplication in A0 [ · 0 ] easily implies that Lx = 0; hence, x# = 0 and x = 0. (ii) We first show that A0 is a left Hilbert algebra in H with involution #. Since the C*-algebra A# has an approximate unit, say {uα }, A0 is dense in the C*-algebra A# and x ≤ x# , for each x ∈ A0 , then it follows that A20 is total in A0 [·0 ]. The assumption, L∗x = Lx # , for all x ∈ A0 , implies that xy|z = y|x # z,

∀ x, y, z ∈ A0 ,

where ·|· is the inner product defined by the Hilbertian norm  · . Moreover, for every x ∈ A0 , the operator Lx is bounded. Take any sequence {xn } in A0 , such that lim xn  = 0 and lim xn # − y = 0, y ∈ A0 . Then, it follows that n→∞

n→∞







y|x1 x2  = lim xn# |x1 x2  = lim x2 x1 |xn  = 0, n→∞

n→∞

∀ x1 , x2 ∈ A0 ,

which implies that the involution map x ∈ A0 → x # ∈ A0 is closable. Thus, A0 is a left Hilbert algebra in H under the involution #. Following the notations of Example 5.2.6, we denote by S the closure of the involution # of A0 , and by J, , the modular conjugation and the modular operator of A0 , respectively. We next show that the full left Hilbert algebra A0 has a unit u. For any ε > 0 and for any finite subsets {x1 , . . . , xm } and {y1 , . . . , ym } of A0 , we define the set

K ε : {x1 , . . . , xm }, {y1 , . . . , ym }  := a ∈ H : a ≤ 1, |axk − xk |yk | ≤ ε

 and |xk a − xk |yk | ≤ ε, k = 1, . . . , m .

unit and x ≤ x# , for Since the C*-algebra A# has an approximate each x ∈ A# , it follows that the set K ε : {x1 , . . . , xm }, {y1 , . . . , ym } is nonempty. Let now K be the family of all such subsets, where ε > 0 and {x1 , . . . , xm }, {y1 , . . . , ym } are finite subsets of A0 . Then, K is a family of nonempty weakly closed subsets of the weakly compact set H1 ≡ {a ∈ H : a ≤ 1}. Hence, the intersection of all the sets in K is non empty. Therefore, an element u of this intersection is such that u is a quasi-unit of the topological quasi *-algebra (A[ · ], A0 ), that is u ∈ A[ · ] and ux = xu = x, for every x ∈ A0 . Since Sx|u = x # |u = u|Lx u = u|x,

∀ x ∈ A0 ,

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it follows that u ∈ D(S ∗ ) and S ∗ u = u. Thus, u ∈ A0 and therefore S ∗ y|u = LS ∗ y u|u = u|Ry u = u|y,

∀ y ∈ A0 .

Consequently, we have u ∈ A0 and Su = u. This completes the proof.

 

By Proposition 5.5.2, the situation of HCQ*-algebras is represented by the following scheme ⊂ A# ⊂ A0  = L(A0 ) u ⊂ D(S) ⊂ A0 " JA

"J

"J

A[ · ].

⊂ A ⊂ A0 = L(A0 ) u ⊂ D(S ∗ ) ⊂ We now look for conditions under which JA = J . Lemma 5.5.3 Let (A[ · ], #) be an HCQ*-algebra. Then, the following statements are equivalent: (i) JA = J ; (ii) x # |x ∗  ≥ 0, for each x ∈ A0 . Proof (i) ⇒ (ii) This follows from 1

1

x # |x ∗  = J  2 x|J x = x| 2 x ≥ 0,

∀ x ∈ A0 . 1

1

(ii) ⇒ (i) By the assumption (ii) we have S = JA (JA J  2 ) and JA J  2 ≥ 0. The uniqueness of the polar decomposition of S implies J = JA .   If anyone of the two equivalent statements of Lemma 5.5.3 holds, we say that the HCQ*-algebra (A[ · ], #) is standard.  0 [ · ] Let (A[ · ], #), (B[ · ], #) be two HCQ*-algebras, with B = B (completion of B0 [ · ]), for some left Hilbert algebra B0 . In general, if A[ · ] = B[ · ] as Hilbert spaces, (A[ · ], #) and (B[ · ], #) need not coincide as HCQ*algebras. For this reason we introduce the following notion: Definition 5.5.4 An HCQ*-algebra (A[ · ], #A ) is said to be an extension of an  0 [ · ], for some left Hilbert algebra B0 ) HCQ*-algebra (B[ · ], #B ) (with B = B if B0 is a dense *-subalgebra of A0 and the involutions #A , #B coincide on B0 . Proposition 5.5.5 Let (A[ · ], #) be a standard HCQ*-algebra and B0 := A00 the maximal Tomita algebra (Example 5.2.6) of the full left Hilbert algebra A0 . Then, (B[ · ], #) is a standard HCQ*-algebra and it is an extension of (A[ · ], #). Further, {it }t∈R is a one-parameter group of *-automorphisms of the Hilbert quasi *-algebra B[·], that is it B0 = B0 , (it a)∗ = it a ∗ , it (ax) = (it a)(it x) and it (xa) = (it x)(it a), for all a ∈ B[ · ], x ∈ B0 and t ∈ R.

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Proof It is almost clear that B[ · ] is a Hilbert quasi *-algebra with the involution JA = JB and further (B[ · ], #A ) is a standard HCQ*-algebra. Since {itA }t∈R is a one-parameter group of *-automorphisms of the Tomita algebra B0 , it follows that {it }t∈R is also a one-parameter group of *-automorphisms of the Hilbert quasi *-algebra B[ · ].   Finally we consider the question of when a Hilbert space can be regarded as a standard HCQ*-algebra. By Propositions 5.5.2, 5.5.5 and [26, Theorem 13.1] we have the following Theorem 5.5.6 Let H be a Hilbert space. The following statements are equivalent: (i) H is a standard HCQ*-algebra; (ii) H contains a left Hilbert algebra with unit as dense subspace; (iii) There exists a von Neumann algebra on H with a cyclic and separating vector. Remark 5.5.7 The implication (iii) ⇒ (i) shows that the class of standard HCQ*algebras is rather rich. Moreover, it gives conditions for the existence of a von Neumann algebra on H with a cyclic and separating vector in the case when H is a nonseparable space. In fact, in the separable case, every von Neumann algebra M is σ -finite; i.e., any collection of mutually orthogonal projections have at most countable cardinality. This is equivalent to the fact that M is isomorphic to a von Neumann algebra possessing a cyclic and separating vector, see [6, Section 2.5.1].

5.5.1 The Structure of Strict CQ*-Algebras In this subsection we study when a strict CQ*-algebra is embedded in a standard HCQ*-algebra (for the definitions of *-homomorphism, *-isomorphism, etc see beginning of Sect. 5.4.). For that, we need a GNS-like construction for a class of positive sesquilinear forms on strict CQ*-algebras (A[ · ], A0 , #) We remind the reader that a positive sesquilinear form ϕ is called faithful if ϕ(a, a) = 0, a ∈ A[ · ], implies a = 0. Theorem 5.5.8 Let (A[·], A0 , #) be a strict CQ*-algebra with quasi-unit u. Then, the following statements are equivalent: (i) there exists a contractive *-homomorphism (resp. *-isomorphism) of the strict CQ*-algebra (A[ · ], A0 , #) into an HCQ*-algebra (B[ ·  ], # ); (ii) there exists a positive (resp. faithful) sesquilinear form ϕ on A × A satisfying the following properties: (ii)1 ϕ(x, y) = ϕ(u, x # y), ∀ x, y ∈ A0 ; (ii)2 |ϕ(x, y)| ≤ xy, ∀ x, y ∈ A0 ; (ii)3 ϕ(x, y) = ϕ(y ∗ , x ∗ ), ∀ x, y ∈ A0 ;

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further, (B[ ·  ], # ) is standard, if and only if, (ii)4 ϕ(x ∗ , x # ) ≥ 0,

∀ x ∈ A0 .

Proof (i) ⇒ (ii) We put ϕ(a, b) =  (a)| (b), a, b ∈ A where ·|· is the inner product defined by the Hilbertian norm  ·  on B. Then, it is easily shown that ϕ is a positive sesquilinear form on A[ · ] × A[ · ] satisfying the conditions (ii)1 –(ii)3 . If (B[ ·  ], # ) is standard, then (ii)4 follows from Lemma 5.5.3. (ii) ⇒ (i) We put Nϕ = {a ∈ A : ϕ(a, a) = 0}. Then, Nϕ is a subspace of A[ · ] and due to the positivity of ϕ, the quotient space λϕ (A) ≡ A/Nϕ = {λϕ (a) ≡ a + Nϕ : a ∈ A} is a pre-Hilbert space with inner product given by λϕ (a)|λϕ (b)ϕ = ϕ(a, b),

∀ a, b ∈ A.

We denote by  · ϕ the norm defined by the inner product ·|·ϕ and by Hϕ the completion of the pre-Hilbert space λϕ (A)[ · ϕ ]. Since A0 is  · -dense in A, it follows that (ii)2 |ϕ(a, b)| ≤ a b, (ii)3 ϕ(a, b) = ϕ(b∗ , a ∗ ),

∀ a, b ∈ A; ∀ a, b ∈ A.

Now by (ii)1 and the inequality x ≤ x# , ∀ x ∈ A0 , we have that (ii)1 ϕ(x, y) = ϕ(u, x # y),

∀ x, y ∈ A# .

By (ii)2 , Aϕ := λϕ (A) is a dense subspace of the Hilbert space Hϕ and moreover, it is a *-algebra equipped with a multiplication defined by λϕ (x)λϕ (y) := Lλϕ (x) λϕ (y) ≡ λϕ (xy),

∀ x, y ∈ Aϕ

and an involution given as follows λϕ (x)∗ := λϕ (x ∗ ),

∀ x ∈ Aϕ .

From (ii)3 , the involution λϕ (x) → λϕ (x)∗ , x ∈ Aϕ , can be extended to an isometric involution Jϕ on Hϕ . From (ii)1 , the linear functional x ∈ A# → ϕ(x, u) ∈ C is positive and so ϕ(y # (x # x)y, u) ≤ x2# ϕ(y, y),

∀ x, y ∈ A0 .

Hence, from (ii)1 , we obtain λϕ (x)λϕ (y)2ϕ = ϕ(xy, xy) = ϕ(y # x # xy, u) ≤ x2# λϕ (y)2ϕ ,

∀ x, y ∈ A0 ,

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so that Lλϕ (x) is bounded and Lλϕ (x)  ≤ x# , for each x ∈ A0 . Thus, Hϕ ≡ ϕ [ · ϕ ] is a Hilbert quasi *-algebra. Further, the map λϕ (x) → λϕ (x)# ≡ λϕ (x # ) A is an involution of Aϕ and by (ii)1 , L∗λϕ (x) = Lλϕ (x)# , for each x ∈ A0 . Hence, ϕ [ · ϕ ], #) is an HCQ*-algebra. (A Now, we put (a) := λϕ (a), a ∈ A. Then, it is easily shown that is a *ϕ [ · ϕ ], #), homomorphism of the strict CQ*-algebra into the HCQ*-algebra (A and by (ii)2 , it is contractive. Suppose that ϕ is faithful. Then, the *-representation of the C*-algebra A# on Hϕ defined by x → Lλϕ (x) , x ∈ A# , is faithful, which implies that Lλϕ (x)  = x# , for each x ∈ A# . Moreover, since (A0 ) = Aϕ , it follows that (A# ) = (Aϕ )# and  A# is a *-isomorphism of the C*-algebra A# onto the C*-algebra (Aϕ )# . Consequently, is a *-isomorphism of (A[ · ], A0 , #) ϕ [ · ϕ ], #). By Lemma 5.5.3, the HCQ*-algebra (A ϕ [ · ϕ ], #) is standard, into (A if and only if, (ii)4 holds. This completes the proof.   Now a question arises as to whether positive sesquilinear forms as described in (ii) do really exist. The answer is certainly positive due to the existence of standard HCQ*-algebras stated in Theorem 5.5.6. Indeed, the inner product ·|· of a left Hilbert algebra satisfies conditions (ii)1 –(ii)4 . In conclusion, Theorem 5.5.8 gives an answer to the main question of this subsection, in the following way: any form ϕ over a strict CQ*-algebra (A[ · ], A0 , #) with quasi-unit, can be used to construct an HCQ*-algebra, in which A[ · ] is contractively embedded.

5.6 Measurable Operators and CQ*-Algebras Another interesting situation where CQ*-algebras play an important role is that of noncommutative integration and in particular that of noncommutative Lp spaces constructed on a von Neumann algebra possessing a normal faithful, semifinite trace.

5.6.1 Noncommutative Measure and Integration Let M be a von Neumann algebra on a Hilbert space H and  a normal faithful semifinite trace defined on M+ . For each p ≥ 1, let   Jp := X ∈ M : (|X|p ) < ∞ .

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127

Then, Jp is a *-ideal of M. Following [70], we denote with Lp () the Banach space completion of Jp with respect to the norm Xp, := (|X|p )1/p ,

X ∈ Jp .

 We shall simply write Xp instead of Xp, , whenever no ambiguity can arise.

One usually defines L∞ () := M. Thus, if  is a finite trace, then L∞ () ⊂ for every p ≥ 1. The definition of noncommutative Lp -spaces given here follows Nelson [71]. Operators in this kind of spaces constructed by a von Neumann algebra M are, however, measurable in Segal sense [76] and then often they are affiliated with M [13]. We give now here an outline of Segal approach. Let P ∈ Proj(M), the lattice of projections of M. Two projections P , Q ∈ Proj(M) are called equivalent and we write P ∼ Q, if there is U ∈ M, such that U ∗ U = P and U U ∗ = Q. We say that P ≺ Q in the case when P is equivalent to a subprojection of Q. A projection P of a von Neumann algebra M is said to be finite if P ∼ Q ≤ P implies P = Q. A projection P ∈ M is said to be purely infinite if there is no nonzero finite projection Q & P in M. A von Neumann algebra M is said to be finite (resp. purely infinite) if the identity operator e is finite (resp. purely infinite). We say that P is -finite if P ∈ J1 . Any -finite projection is finite. In what follows we shall need the following result (see, [64, Vol. IV, Ex. 6.9.12]) that we state as a lemma. Lp (),

Lemma 5.6.1 Let M be a von Neumann algebra on a Hilbert space H and  a normal, faithful, semifinite trace on M+ . Then, there is# an orthogonal family {Qj : j ∈ J } of nonzero central projections in M, such that j ∈J Qj = e and each Qj is the sum of an orthogonal family of mutually equivalent finite projections in M. A vector subspace D of H is said to be strongly dense (resp. strongly -dense) if • U  D ⊂ D, for any unitary operator U  in M ; • there exists a sequence {Pn } ∈ Proj(M) : Pn H ⊂ D, Pn⊥ ↓ 0 and Pn⊥ is a finite projection (resp. (Pn⊥ ) < ∞). Clearly, every strongly -dense domain is strongly dense. Throughout this subsection, when we say that an operator T is affiliated with a von Neumann algebra M and write T η M, we always mean that T is closed, densely defined and T U ⊇ U T , for every unitary operator U ∈ M . An operator T η M is called • measurable (with respect to M) if its domain D(T ) is strongly dense; • -measurable if its domain D(T ) is strongly -dense. From the definition itself it follows that, if T is -measurable, then there exists P ∈ Proj(M), such that T P is bounded and (P ⊥ ) < ∞.

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5 CQ*-Algebras

Remark 5.6.2 As shown in [70], every operator X ∈ Lp () is measurable. Moreover, every operator affiliated with a finite von Neumann algebra is measurable, but not necessarily -measurable (see [76, Cor. 4.1]). In the general case, conditions that guarantee the measurability of affiliated operators are discussed in [27, IX, §2]. If A is a measurable operator and A ≥ 0, one defines the integral of A by   μ(A) = sup (X) : 0 ≤ X ≤ A, X ∈ J1 . Then, the space Lp () can also be defined [70] as the space of all measurable operators A, such that μ(|A|p ) < ∞. The integral of an element A ∈ Lp () can be defined, in obvious way, taking into account that any measurable operator A can be decomposed as A = B+ − B− + ∗ A−A∗ iC+ − iC− , where B = A+A and B+ , B− (resp. C+ , C− ) are the 2 , C = 2i positive and negative parts of B (resp. C). Remark 5.6.3 The following statements will be used later:   (i) Let T η M and Q ∈ M. If D(T Q) = ξ ∈ H : Qξ ∈ D(T ) is dense in H, then T Q η M.   (ii) If Q ∈ Proj(M), then QMQ = QSQ QH : S ∈ M is a von Neumann algebra on the Hilbert space QH; moreover, (QMQ) = QM Q. If T η M, Q ∈ M and D(T Q) = ξ ∈ H : Qξ ∈ D(T ) is dense in H, then QT Q η QMQ. For (i) and (ii), see for instance, [14, Chapter 5].

5.6.2 Noncommutative Lp -Spaces as Proper CQ*-Algebras In this subsection we shall discuss the structure of the noncommutative Lp -spaces as quasi *-algebras. Proposition 5.6.4 Let M be a von Neumann algebra and  a normal faithful semifinite trace on M+ . Then, (Lp (), L∞ () ∩ Lp ()) is a Banach quasi *algebra. If  is a finite trace and (I) = 1, then (Lp (), L∞ ()) is a proper CQ*-algebra. Proof Indeed, it is easily seen that the norms  · ∞ on L∞ () ∩ Lp () and  · p on Lp () satisfy the conditions of Definition 3.1.1. Moreover, if  is finite, then L∞ () ⊂ Lp () and thus (Lp (), L∞ ()) is a proper CQ*-algebra.   Remark 5.6.5 Of course, the condition (I) = 1 can be easily removed by normalizing the trace. We shall now focus our attention on the question as to whether for the Banach quasi *-algebra (Lp (), L∞ () ∩ Lp ()), the family SL∞ () (Lp ()), that we are going to describe, is or is not sufficient.

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129

Before going forth, we remind that many of the familiar results of the ordinary theory of Lp -spaces hold in the very same form for the noncommutative Lp -spaces. This is the case, for instance, of Hölder’s inequality and also of the statement that characterizes the dual of Lp : the form defining the duality is the extension of  (this extension will be denoted with the same symbol) to products of the type XY with  X ∈ Lp (), Y ∈ Lp () and p−1 + p −1 = 1. Moreover, one has (Lp ())   Lp (). In order to study SL∞ () (Lp ()), we introduce, for p ≥ 2, the following notation p   p B+ := X ∈ L p−2 (), X ≥ 0 , Xp/(p−2) ≤ 1

where p/(p − 2) = ∞ if p = 2. p For each W ∈ B+ , we consider the right multiplication operator p

RW : Lp () → L p−2 () : X → RW X := XW, X ∈ Lp (). Since L∞ () ∩ Lp () = Jp , we use, for shortness, the latter notation. Lemma 5.6.6 Let p ≥ 2. The following statements hold:

p ∗ (i) for every W ∈ B +p, the sesquilinear form ϕ(X, Y ) :=  X(RW Y ) is an element of SL∞ () L () ;

p (ii) if  is finite, then for each ϕ ∈ SL∞ () Lp () , there exists W ∈ B+ , such that

ϕ(X, Y ) =  X(RW Y )∗ ,

∀ X, Y ∈ Lp ().

Proof



(i) We check that the sesquilinear form, ϕ(X, Y ) =  X(RW Y )∗ , X, Y ∈ Lp (), satisfies the conditions of Sect. 3.1.2. For every X ∈ Lp (), we have



ϕ(X, X) =  X(RW X)∗ =  (XW )∗ X =  W |X|2 ≥ 0. For every X ∈ Lp (), A, B ∈ Jp , we get

ϕ(XA, B) =  XA(BW )∗ = (W B ∗ XA) =  A(X∗ BW )∗ = ϕ(A, X∗ B). Finally, for every X, Y ∈ Lp (), |ϕ(X, Y )| ≤ Xp Y p W p/p−2 ≤ Xp Y p .

 (ii) Let ϕ ∈ SL∞ () Lp () . Let T : Lp () → Lp () be the operator which represents ϕ in the sense of Proposition 3.1.16. The finiteness of  implies that Jp = M; thus we can put W = T (I). It is easy to check that RW = T . This concludes the proof.  

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Proposition 5.6.7 If p ≥ 2, SL∞ () Lp () is sufficient. Proof Let X ∈ Lp () be such that ϕ(X, X) = 0, for every ϕ ∈ S(Lp ()). By p

the previous lemma, since |X|p−2 ∈ L p−2 (), the right multiplication operator p−2 p−2 RW with W = |X|α , α ∈ R \ {0}, satisfying  |X|α p/p−2 ≤ 1, represents a sesquilinear form ϕ ∈ SL∞ () (Lp ()). By the assumption, ϕ(X, X) = 0. We then have

ϕ(X, X) =  X(RW X)∗ = α1  X(X|X|p−2 )∗

= α1  (X|X|p−2 )∗ X = α1 (|X|p ) = 0 ⇒ X = 0,  

by the faithfulness of .

5.6.3 CQ*-Algebras over Finite von Neumann Algebras Let M be a von Neumann algebra and F = {α ; α ∈ I} be a family of normal, finite traces on M. As usual, we say that the family F is sufficient if, for X ∈ M, X ≥ 0 and α (X) = 0, for every α ∈ I, then X = 0 (clearly, if F = {}, then F is sufficient, if and only if,  is faithful). In this case, M is a finite von Neumann algebra [25, ch.7]. We assume, in addition, that the following condition (5.6.17) is satisfied: α (I) ≤ 1,

∀ α ∈ I,

(5.6.17)

where I is the identity of M. Then, we define Xp,I = sup Xp,α = sup α (|X|p )1/p . α∈I

α∈I

Since F is sufficient,  · p,I is a norm on M. In the sequel we shall need the following lemmas, whose simple proofs will be omitted. Lemma 5.6.8 Let M be a von Neumann algebra in a Hilbert space H and {Pα }α∈I a family of projections of M with $

Pα = P .

α∈I

If A ∈ M and APα = 0, for every α ∈ I, then AP = 0. Lemma 5.6.9 Let F = {α }α∈I be a sufficient family of normal, finite # traces on the von Neumann algebra M and let Pα be the support of α . Then, α∈I Pα = I, where I denotes the identity of M.

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131

It is well-known that the support of each α enjoys the following properties: (i) Pα ∈ Z(M), the center of M, for each α ∈ I; (ii) α (X) = α (XPα ), for each α ∈ I. From the two preceding lemmas it follows that, if the Pα ’s are as in Lemma 5.6.9, then APα = 0,

∀α∈I

⇒

A = 0.

If condition (5.6.17) is fulfilled, then Xp,I := sup XPα p,α , α∈I

∀ X ∈ M.

Clearly, the sufficiency of the family of traces and condition (5.6.17) imply that  · p,I is a norm on M. Proposition 5.6.10 Let M(p, I) denote the Banach space completion of M with respect to the norm ·p,I . Then, (M(p, I)[·p,I ], M) is a proper CQ*-algebra. Proof Indeed, from Definition 5.1.1, ∀ X ∈ M, we have X∗ p,I = sup X∗ Pα p,α = sup (XPα )∗ p,α = Xp,I . α∈I

α∈I

(5.6.18)

Furthermore, for every X, Y ∈ M, XY p,I = sup XY Pα p,α ≤ XB(H) sup Y Pα p,α α∈I

α∈I

(5.6.19)

= XB(H) Y p,I . Finally, condition (5.6.17) implies that Xp,I ≤ X,

∀ X ∈ M.

It follows from (5.6.18) and (5.6.19) that (M(p, I)[ · p,I ], M) is a Banach quasi *-algebra and therefore a proper CQ*-algebra.   • The next step aims at investigating the Banach space M(p, I)[ · p,I ]. In particular, we are interested in the question as to whether M(p, I)[ · p,I ] can be identified with a space of operators affiliated with M. For shortness, whenever no ambiguity can arise, we write Mp instead of M(p, I) Let F = {α }α∈I be a sufficient family of normal, finite traces on the von Neumann algebra M satisfying condition (5.6.17). The traces α are not necessarily faithful. Put Mα := MPα , where, as before, Pα denotes the support of α . Each Mα is a von Neumann algebra and α is faithful in MPα [27, Proposition V. 2.10].

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More precisely,   Mα := MPα = Z = XPα , for some X ∈ M . The positive cone M+ α of Mα equals the set 

 Z = XPα , for some X ∈ M+ .

For Z = XPα ∈ M+ α , we put σα (Z) := α (XPα ). The definition of σα (Z) does not depend on the particular choice of X. Each σα is a normal, finite, faithful trace on Mα . It is then possible to consider the spaces Lp (Mα , σα ), p ≥ 1, in the usual way. Recall that the norm of Lp (Mα , σα ) is indicated by  · p,σα . Let now {Xk } be a Cauchy sequence in M[ · p,I ]. For each (α) (α) α ∈ I, we put Zk = Xk Pα . Then, for each α ∈ I, {Zk } is a Cauchy sequence in (α) (α) Mα [ · p,σα ]. Indeed, since |Zk − Zh |p = |Xk − Xh |p Pα , we have

1/p Zk(α) − Zh(α) p,σα = σα |Zk(α) − Zh(α) |p

1/p = α |Xk − Xh |p Pα

1/p = α |Xk − Xh |p → 0. Therefore, for each α ∈ I, there exists an operator Z (α) ∈ Lp (Mα , σα ), such that (α)

Z (α) =  · p,σα − lim Zk . k→∞

It is now natural to ask the question as to whether there exists an operator X closed, densely defined, affiliated with M, which reduces to Z (α) on Mα . To begin with, we assume that the projections {Pα } are mutually orthogonal. In this case, setting Hα = Pα H, we have H=

 α∈I

   Hα = (fα )α∈I : fα ∈ Hα , fα 2 < ∞ . α∈I

We now set    Z (α) fα 2 < ∞ D(X) = (fα )α∈I ∈ H : fα ∈ D(Z (α) ), α∈I

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133

and for f = (fα )α∈I ∈ D(X) we define Xf := (Z (α) fα ). Then, 1. D(X) is dense in H. Indeed, D(X) contains all f = (fα )α∈I , with fα = 0, for all α ∈ I, except for a finite subset of indices. 2. X is closed in H. Indeed, let {fn = (fn,α )α∈I } be a sequence of elements in D(X), with fn → g = (gα )α∈I ∈ H and Xfn → h. Since, fn → g ⇔ fn,α → gα ∈ Hα ,

∀α∈I

and Xfn → h ⇔ (Xfn )α → hα ∈ Hα ,

∀ α ∈ I,

by (Xfn )α = Z (α) fn,α and the closedness of each Z (α) in Hα , we obtain gα ∈ D(Z (α) )

and

hα = Z (α) gα .

 It remains to check that α∈I Z (α) gα 2 < ∞, but this is clear, since both (Z (α) gα )α∈I and h = (hα )α∈I belong to H. 3. X η M. Let Y ∈ M . Then, for all f ∈ H, Yf = (Y Pα f )α∈I and Y Pα ∈ (MPα ) = M Pα , so that

XYf = (XY )Pα f α∈I = (Y XPα f )α∈I = Y Xf. In conclusion, X is a measurable operator. Thus, we have proved the following Proposition 5.6.11 Let F = {α }α∈I be a sufficient family of normal, finite traces on the von Neumann algebra M. Assume that condition (5.6.17) is fulfilled (i.e., α (I) ≤ 1, ∀ α ∈ I) and that the α ’s have mutually orthogonal supports. Then, Mp , p ≥ 1, consists of measurable operators. The analysis of the general case would really be simplified if, from a given sufficient family F of normal, finite traces, one could extract (or construct) a sufficient subfamily G of traces with mutually orthogonal supports. Apart from quite simple situations (for instance, when F is finite or countable), we do not know if this is possible or not. There is however a relevant case, where this can be fairly easily done. This occurs when F is a convex and w*-compact family of traces on M. Lemma 5.6.12 Let F be a convex w ∗ -compact family of normal, finite traces on a von Neumann algebra M. Assume that, for each central operator Z, with 0 ≤ Z ≤ I, and each η ∈ F, the functional ηZ (X) := η(XZ) belongs to F. Let EF be the set of all extreme elements of F. If η1 , η2 ∈ EF, η1 = n2 , and P1 and P2 are their respective supports, then P1 and P2 are orthogonal.

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5 CQ*-Algebras

Proof Let P1 , P2 be the supports of η1 and η2 , respectively. We begin proving that either P1 = P2 or P1 P2 = 0. Indeed, assume that P1 P2 = 0. We define η1,2 (X) := η1 (XP2 ),

∀ X ∈ M.

When η1,2 = 0, then in particular, η1,2 (P2 ) = 0, i.e., η1 (P2 ) = 0 and therefore, by definition of support, P2 ≤ I − P1 . This implies that P1 P2 = 0, which contradicts the assumption. We now show that the support of η1,2 is P1 P2 . In fact, let Q be a projection, such that η1,2 (Q) = 0. Then, η1 (QP2 ) = 0 ⇒ QP2 ≤ I − P1 ⇒ QP2 (I − P1 ) = QP2 ⇒ QP2 P1 = 0. Then, the largest Q for which this happens is I−P2 P1 . We conclude that the support of the trace η1,2 is P1 P2 . Finally, by definition, one has η1,2 (X) = η1 (XP2 ), and since XP2 ≤ X, η1,2 (X) = η1 (XP2 ) ≤ η1 (X),

∀ X ∈ M.

Thus, η1 majorizes η1,2 . But η1 is extreme in F. Therefore, η1,2 has the form λη1 with λ ∈]0, 1]. This implies that η1,2 has the same support as η1 ; hence, P1 P2 = P1 , i.e., P1 ≤ P2 . Starting from η2,1 (X) = η2 (XP1 ), we obtain in a similar way that P2 ≤ P1 . Consequently, P1 P2 = 0 implies P1 = P2 . However, two different traces of EF cannot have the same support. Indeed, assume that there exist η1 , η2 ∈ F having the same support P . Since P is central, we can consider the von Neumann algebra MP . The restrictions of η1 , η2 to MP are normal, faithful, semifinite traces. By [27, Prop. V.2.31] there exist a central element Z in MP with 0 ≤ Z ≤ P (P is here considered as the unit of MP ), such that η1 (X) = (η1 + η2 )(ZX),

∀ X ∈ (MP )+ .

(5.6.20)

Then, Z also belongs to the center of M, since for every V ∈ M, ZV = Z(V P + V P ⊥ ) = ZV P = V ZP = V Z. Therefore, the functionals η1,Z (X) := η1 (XZ),

η2,Z (X) := η2 (XZ),

∀ X ∈ M,

belong to the family F and are majorized respectively, by the extreme elements η1 , η2 . Then, there exist λ, μ ∈ [0, 1], such that η1 (XZ) = λη1 (X),

η2 (XZ) = μη1 (X),

∀ X ∈ M.

If λ = 1, from (5.6.20), we would have η2 (ZX) = 0, for every X ∈ (MP )+ ; in particular, η2 (| Z |2 ) = 0; this implies that Z = 0. Thus, λ = 1. Analogously,

5.6 Measurable Operators and CQ*-Algebras

135

μ = 0; indeed, if μ = 0, then η1 (X) = λη1 (X) and so λ = 1. Therefore, there exist λ, μ ∈ (0, 1), such that η1 (X) = λη1 (X) + μη2 (X),

∀ X ∈ MP ,

which in turn, implies η1 (X) = λη1 (X) + μη2 (X),

∀ X ∈ M.

Hence, (1 − λ)η1 (X) = μη2 (X),

∀ X ∈ M.

From the last equality, dividing by max{1 − λ, μ} one gets that one of the two elements is a convex combination of the other and of 0, which is absurd. In conclusion, different supports of extreme traces of F are orthogonal.   Since, for every X ∈ M, Xp,I remains the same if computed either with respect to F or to EF, we can deduce the following Theorem 5.6.13 Let F be a convex and w ∗ -compact sufficient family of normal, finite traces on the von Neumann algebra M. Assume that F satisfies condition (5.6.17) and that for each central operator Z, with 0 ≤ Z ≤ I, and each η ∈ F, the functional ηZ (X) := η(XZ) belongs to F. Then, Mp [ · p,I ], consists of measurable operators. Families of traces satisfying the assumptions of Theorem 5.6.13 will be constructed in the next subsection.

5.6.4 A First Representation Theorem Once we have constructed in the previous subsection some CQ*-algebras of operators affiliated to a given von Neumann algebra, it is natural to pose the question, under which conditions, can an abstract CQ*-algebra (A[ · ], A0 ) be realized as a CQ*-algebra of operators. Let (A[ · ], A0 ) be a proper CQ*-algebra with unit e and let   TA0 (A) ≡ ϕ ∈ SA0 (A) : ϕ(a, a) = ϕ(a ∗ , a ∗ ), ∀ a ∈ A . Note that if ϕ ∈ TA0 (A), then by polarization, ϕ(b∗ , a ∗ ) = ϕ(a, b), ∀ a, b ∈ A. It is easy to prove that the set TA0 (A) is convex. For each ϕ ∈ TA0 (A), we define a linear functional ωϕ on A0 by ωϕ (x) := ϕ(x, e),

∀ x ∈ A0 .

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5 CQ*-Algebras

For every x ∈ A0 , we have ωϕ (x ∗ x) = ϕ(x ∗ x, e) = ϕ(x, x) = ϕ(x ∗ , x ∗ ) = ωϕ (xx ∗ ) ≥ 0. This shows at once that ωϕ is positive and tracial. We put   MT (A0 ) ≡ ωϕ : ϕ ∈ TA0 (A) . From the convexity of TA0 (A) it follows easily that MT (A0 ) is convex too. Then, for the norm of the bounded linear functional ωϕ , recalling that we may suppose that e = 1 (Remark 3.1.4), we have ωϕ  = ωϕ (e) = ϕ(e, e) ≤ 1. Therefore,   MT (A0 ) ⊆ ω ∈ A0 : ω ≤ 1 , where A0 denotes the topological dual of A0 [ · 0 ]. Setting fϕ (x) := ωϕ (x), x ∈ A0 , we obtain   fϕ ∈ ω ∈ A0 : ω ≤ 1 .   By the Banach–Alaoglou theorem, the set ω ∈ A0 : ω ≤ 1 is a w*-compact subset of A0 . Proposition 5.6.14 MT (A0 ) is w*-closed and, therefore, a w*-compact set. Proof Let (ωϕα ) be a net in MT (A0 ) w*-converging to a functional ω ∈ A0 . We shall show that ω = ωϕ , for some ϕ ∈ TA0 (A). Let us begin defining ϕ0 (x, y) := ω(y ∗ x), x, y ∈ A0 . By the definition itself, (ωϕα )(x) → ω(x) = ϕ0 (x, e). Moreover, for every x, y ∈ A0 , ϕ0 (x, y) = ω(y ∗ x) = lim ωϕα (y ∗ x) = lim ϕα (x, y). α

α

Therefore, ϕ0 (x, x) = lim ϕα (x, x) ≥ 0. α

5.6 Measurable Operators and CQ*-Algebras

137

We also have | ϕ0 (x, y) | = lim | ϕα (x, y) | ≤ x y. α

Hence, ϕ0 can be extended by continuity to A × A. Indeed, let a, b ∈ A such that a =  ·  − lim xn , n

b =  ·  − lim yn , with {xn }, {yn } ⊂ A0 . n

Then, | ϕ0 (xn , yn ) − ϕ0 (xm , ym ) | = | ϕ0 (xn , yn ) − ϕ0 (xm , yn ) + ϕ0 (xm , yn ) − ϕ0 (xm , ym )| ≤ | ϕ0 (xn − xm , yn ) | + | ϕ0 (xm , yn − ym ) | ≤ xn − xm  yn  + xm  yn − ym  → 0, since {xn } and {yn } are bounded sequences. Hence, we can define ϕ(a, b) := lim ϕ0 (xn , yn ). n

Clearly, ϕ(a, a) ≥ 0, for all a ∈ A. It is then easily checked that finally ϕ ∈ TA0 (A). This completes the proof.

 

Since MT (A0 ) is convex and w*-compact, by the Krein–Milmann theorem it follows that it has extreme points and it coincides with the w*-closure of the convex hull of the set EMT (A0 ) of its extreme points. Let π be the universal *-representation of the C*-algebra A0 (Remark A.6.16)  and π(A0 ) the von Neumann algebra generated by π(A0 ). For every ϕ ∈ TA0 (A) and x ∈ A0 , we put ϕ (π(x)) := ωϕ (x). Then, for each ϕ ∈ TA0 (A), ϕ is a positive bounded linear functional on the operator algebra π(A0 ). Clearly, ϕ (π(x)) = ωϕ (x) = ϕ(x, e),

∀ x ∈ A0 and

| ϕ (π(x)) | = | ωϕ (x) | = | ϕ(x, e) |≤ x ≤ x0 = π(x) ,

∀ x ∈ A0 .

Thus, ϕ is continuous on π(A0 ) and then according to [15, Proposition 10.1.1],  ϕ is also weakly continuous and so it extends uniquely to π(A0 ) . Moreover, since ϕ is a trace on M := π(A0 ) too. The norm ϕ is a trace on π(A0 ), the extension   ϕ on M equals the norm of the linear functional ϕ ϕ  of the linear functional   on π(A0 ). Moreover, ϕ  =  ϕ (π(e)) = ϕ (π(e)) = ωϕ (e) ≤ 1. 

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5 CQ*-Algebras

The set   ϕ : ϕ ∈ TA0 (A) NT (A0 ) ≡  is convex and w*-compact in M , as it can be easily seen by considering the map ϕ ∈ NT (A0 ), ωϕ ∈ MT (A0 ) →  which is linear and injective and taking into account the fact that, if xn → x in ϕ (π(xn ) − π(x)) = ωϕ (xn − x) → 0. A0 [ · ], then  Let ENT (A0 ) be the set of the extreme points of NT (A0 ); then NT (A0 ) coincides with the w*-closure of the convex hull of ENT (A0 ). The extreme elements of NT (A0 ) are easily characterized by the following ϕ is extreme in NT (A0 ), if and only if, ωϕ is extreme in Proposition 5.6.15  MT (A0 ). Definition 5.6.16 A Banach quasi *-algebra (A[ · ], A0 ) is said to be strongly regular if TA0 (A) is sufficient and x =

sup

ϕ∈TA0 (A)

ϕ(x, x)1/2 ,

∀ x ∈ A.

Example 5.6.17 If M is a von Neumann algebra possessing a sufficient family F of normal, finite traces, then the proper CQ*-algebra (Mp , M) constructed in the previous subsection is strongly regular. This follows from the definition itself in the completion. Example 5.6.18 If  is a normal, faithful, finite trace on M, then TJp (Lp ()), for p ≥ 2, is sufficient. To see this, we start defining ϕ0 on M × M by ϕ0 (X, Y ) := (Y ∗ X),

∀ X, Y ∈ M.

Then, |ϕ0 (X, Y )| = |(Y ∗ X)| ≤ Xp Y p ,

∀ X, Y ∈ M,

where p is the conjugate of p. Since p ≥ 2, then Lp () is continuously embedded  into Lp (). Thus, there exists γ > 0, such that Y p ≤ γ Y p , for every Y ∈ M. Let us define  ϕ (X, Y ) :=

1 ϕ0 (X, Y ), γ

∀ X, Y ∈ M.

5.6 Measurable Operators and CQ*-Algebras

139

Then, | ϕ (X, Y )| ≤ Xp Y p ,

∀ X, Y ∈ M.

Hence,  ϕ has a unique extension to Lp () × Lp (), denoted with the same symbol. It is easily seen that  ϕ ∈ TJp (Lp ()). Assume that there exists X ∈ Lp (), such that ϕ(X, X) = 0, for every ϕ ∈ TJp (Lp ()), then,  ϕ (X, X) = X22 = 0. This clearly implies X = 0. The equality 2  ϕ (X, X) = X2 also shows that L2 () is strongly regular. Let now (A[ · ], A0 ) be a proper CQ*-algebra with unit e and TJp (A) sufficient. Let π : A0 → B(H) be the universal representation of A0 . Assume that the C*-algebra π(A0 ) := M is a von Neumann algebra. In this case, MT (A0 ) = NT (A0 ) and NT (A0 ) is a family of traces satisfying condition (5.6.17). Therefore, by Proposition 5.6.10, we can construct for p ≥ 1, the proper CQ*-algebras (Mp [·p, NT (A0 ) ], M[·]). Clearly, A0 can be identified with M. It is then natural to pose the question if also A can be identified with some Mp . The next Theorem provides an answer to this question. Theorem 5.6.19 Let (A[ · ], A0 ) be a proper CQ*-algebra with unit e, such that TA0 (A) is sufficient. Then, there exist a von Neumann algebra M and a monomorphism  ∈ M2 ,

: a ∈ A → (a) := X with the following properties: (i) extends the universal *-representation π of A0 ; (ii) (a ∗ ) = (a)∗ , ∀ a ∈ A; (iii) (ab) = (a) (b), ∀ a, b ∈ A, such that a ∈ A0 or b ∈ A0 . Then, A can be identified with a space of operators affiliated with M. If, in addition, (A[ · ], A0 ) is strongly regular, then (iv) is an isometry of A into M2 ; (v) if A0 is a W*-algebra, then is an isometric *-isomorphism of A onto M2 . Proof Let π be the universal representation of A0 and assume first that π(A0 ) =: M is a von Neumann algebra. By Proposition 5.6.14, the family of traces MT (A0 ) is convex and w*-compact. Moreover, for each central positive element Z ∈ M with 0 ≤ Z ≤ I and for ω ∈ MT (A0 ), the trace ωZ (X) := ω(ZX) yet belongs to MT (A0 ). Indeed, starting from the form ϕ ∈ TA0 (A), such that ω = ωϕ , one can define the sesquilinear form

ϕZ (a, b) := ϕ aπ −1 (Z 1/2 ), bπ −1 (Z 1/2 ) ,

∀ a, b ∈ A.

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5 CQ*-Algebras

We check that ϕZ ∈ TA0 (A).

(i) ϕZ (a, a) = ϕ aπ −1 (Z 1/2 ), aπ −1 (Z 1/2 ) ≥ 0, (ii) For every a ∈ A and for every x, y ∈ A0 ,

∀ a ∈ A.



ϕZ (ax, y) = ϕ axπ −1 (Z 1/2 ), yπ −1 (Z 1/2 )

= ϕ xπ −1 (Z 1/2 ), a ∗ yπ −1 (Z 1/2 ) = ϕZ (x, a ∗ y). (iii) For every a, b ∈ A,

| ϕZ (a, b) | = | ϕ aπ −1 (Z 1/2 ), bπ −1 (Z 1/2 ) | ≤ aπ −1 (Z 1/2 ) bπ −1 (Z 1/2 )b ≤ a π −1 (Z 1/2 )0 b π −1 (Z 1/2 )0 ≤ a b. The latter inequality follows from the C*-condition π −1 (Z 1/2 )20 π −1 (Z)0 and the fact that π −1 (Z)0 = Z ≤ 1. (iv) For every a ∈ A,

=



ϕZ (a ∗ , a ∗ ) = ϕ a ∗ π −1 (Z 1/2 ), a ∗ π −1 (Z 1/2 )

= ϕ aπ −1 (Z 1/2 ), aπ −1 (Z 1/2 ) = ϕZ (a, a). Moreover, for every A = π(x) ∈ M = π(A0 ), ϕZ defines the following trace

φϕZ (A) = ϕZ (x, e) = ϕ xπ −1 (Z 1/2 ), π −1 (Z 1/2 )



= ϕ xπ −1 (Z), e = ϕ π −1 (AZ), e = φϕ (AZ). Then, the family of traces NT (A0 ) (= MT (A0 )) satisfies the assumptions of Lemma 5.6.12; therefore, if η1 , η2 ∈ ENT (A0 ), denoting with P1 and P2 their respective supports, one has P1 P2 = 0. By the sufficiency of TA0 (A) we get X2,MT (A0 ) :=

sup ϕ∈MT (A0 )

X2,ϕ =

sup

X2,ϕ ,

∀ X ∈ π(A0 ).

ϕ∈EMT (A0 )

By Proposition 5.6.10, the Banach space M2 , completion of M with respect to the norm  · 2, NT (A0 ) , is a proper CQ*-algebra. Moreover, since the supports of the extreme traces satisfy the assumptions of Theorem 5.6.13, the proper CQ*-algebra (M2 [ · 2,NT (A0 ) ], M[ · ]), consists of operators affiliated with M. We shall now define the map .

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141

For every element a ∈ A, there exists a sequence {xn } of elements of A0 converging to a with respect to  · . Put Xn = π(xn ), n ∈ N. Then, Xn − Xm 2,NT (A0 ) := = =

sup

π(xn ) − π(xm )2,ϕ

sup



1/2 ϕ (xn − xm )∗ (xn − xm ), e

sup

ϕ(xn − xm , xn − xm )1/2 ≤ xn − xm  → 0.

ϕ∈TA0 (A)

ϕ∈TA0 (A) ϕ∈TA0 (A)

 be the  · 2,M (A ) -limit of the sequence {Xn } in M2 . We define (a) := X.  Let X 0 T For each a ∈ A, we put pTA0 (A) (a) =

sup

ϕ∈TA0 (A)

ϕ(a, a)1/2 .

Then, due to the sufficiency of TA0 (A), pTA (A) is a norm on A weaker than  · . 0 This implies that  22,N (A ) = lim X 0 T

sup

n→∞ ϕ∈T

A0 (A)

ϕ(xn , xn ) = lim pTA n→∞

0

(A)

(xn )2 = pTA

0

(A)

(a)2 .

From this equality it follows easily that the linear map is well defined and injective. The condition (iii) can be easily proved. If (A[ · ], A0 ) is strongly regular, then pTA (A) (a) = a, for every a ∈ A. Thus, is isometric. Moreover, 0 in this case, is surjective; indeed, if T ∈ M2 , then there exists a sequence {Tn } of bounded operators in π(A0 ), which converges to T , with respect to the norm  · 2,NT (A0 ) . The corresponding sequence, say {yn } in A0 with Tn = (yn ), converges to some b in A, with respect to the norm of A and (b) = T , by definition. Therefore, is an isometric *-isomorphism. To complete the proof, it is enough to show that the given proper CQ*-algebra (A[ · ], A0 ) can be embedded in a proper CQ*-algebra (B[ · 2,NT (A0 ) ], B0 ), where B0 is a W*-algebra. Of course, we may directly work with π(A0 ), where π is the universal representation of A0 . The family of traces NT (A0 ) defined on π(A0 ) ϕ and let is not necessarily sufficient. Let Pϕ , ϕ ∈ TA0 (A), denote the support of  $

P :=

Pϕ .

ϕ∈TA0 (A)

Then, B0 := π(A0 ) P is a von Neumann algebra, that we can complete, with respect to the norm X2,NT (A0 ) =

sup

ϕ∈TA0 (A)

ϕ (X∗ X), 

X ∈ π(A0 ) P .

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5 CQ*-Algebras

We obtain, in this way, a proper CQ*-algebra (B[ · 2,NT (A0 ) ], B0 ) with B ≡ 0 ·2,NT (A0 ) (i.e.,  · 2,N (A ) -completion of B0 ) and B0 a W*-algebra. The B 0 T faithfullness of π on A0 implies that π(x)P = π(x),

∀ x ∈ A0 .

It remains to prove that A can be identified with a subspace of K. But this can be shown in the very same way as we did in the first part of the proof, concerning definition of ; i.e., for each a ∈ A there exists a sequence {xn } in A0 , such that a − xn  → 0, as n → ∞. We now put Xn = π(xn ). Then, proceeding as before,  ∈ B, where we determine the element X  =  · 2,N (A ) − lim π(xn )P . X 0 T  ∈ B is injective. If (A[ · ], A0 ) is It is easy to see that the map a ∈ A → X  strongly regular, but π(A0 ) ⊂ π(A0 ) , then is an isometry of A into M2 , that it needs not be surjective.  

Chapter 6

Locally Convex Quasi *-Algebras

This chapter is devoted to locally convex quasi *-algebras and locally convex quasi C*-algebras. Both these notions generalize what we have discussed in Chaps. 3 and 5. The advantage is, of course, that the range of applications becomes larger and larger; the drawback is that the theory becomes more involved.

6.1 Representable Functionals on Locally Convex Quasi *-Algebras Definition 6.1.1 Let (A, A0 ) be a quasi *-algebra and τ a locally convex topology on A. We say that (A[τ ], A0 ) is a locally convex quasi *-algebra if (i) the map a ∈ A → a ∗ ∈ A is continuous; (ii) for every x ∈ A0 , the maps a → ax, a → xa from A[τ ] into A[τ ] are continuous; (iii) A0 is τ -dense in A. Example 6.1.2 If L† (D, H) is endowed with the strong* topology ts ∗ (see beginning of Sect. 2.1.3), defined by the family of seminorms pξ (X) := Xξ  + X† ξ ,

ξ ∈ D, X ∈ L† (D, H),

then (L† (D, H)[ts ∗ ], L† (D)b ) is a locally convex quasi *-algebra. Recall that L† (D)b is the bounded part of L† (D) (ibid.). Indeed, the continuity of the involution X → X† , X ∈ L† (D, H), comes immediately from the definition of pξ . Moreover, if ξ ∈ D, X ∈ L† (D, H) and Y ∈ L† (D)b , the obvious inequality XY ξ  + Y † X† ξ  ≤ XY ξ  + X† Y ξ  + Y (Xξ  + X† ξ ) © Springer Nature Switzerland AG 2020 M. Fragoulopoulou, C. Trapani, Locally Convex Quasi *-Algebras and their Representations, Lecture Notes in Mathematics 2257, https://doi.org/10.1007/978-3-030-37705-2_6

143

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implies the continuity of X → XY (and of X → Y X, by applying involution). Moreover, L† (D)b is dense in L† (D, H)[ts ∗ ] by [2, Example 2.5.10]. If L† (D, H) is endowed with the weak topology tw defined by the set of seminorms pξ,η (X) := |Xξ |η|,

ξ, η ∈ D, X ∈ L† (D, H),

then, again, (L† (D, H)[tw ], L† (D)b ) is a locally convex quasi *-algebra. Example 6.1.3 As proved in [45, Proposition 6.2], (L† (D, H)[ts ∗ ], L† (D)b ) exhibits a richer structure than that of a locally convex quasi *-algebra. It is, indeed, a locally convex C*-normed algebra, whose definition we discuss here (for more details, see [45] and Sect. 7.1). Let A0 [ · 0 ] be a unital C*-normed algebra and τ a locally convex *-algebra topology on A0 , with {pλ }λ∈ a defining family of *-seminorms. Suppose that the following conditions are satisfied: (T1 ) A0 [τ ] is a locally convex ∗-algebra with separately continuous multiplication. (T2 ) τ &  · 0 , with τ and  · 0 being compatible (in the sense that for any Cauchy net {xα } in A0 [ · 0 ], such that xα → 0 in τ , xα → 0 in  · 0 ). 0 [τ ] denotes the completion of the C*-normed algebra A0 with respect Then, if A to τ , we conclude from (T2 ) and (T1 ), respectively that 0 [ · 0 ] → A 0 [τ ]; and • A0 [ · 0 ] → A  • A0 [τ ] is a locally convex quasi *-algebra over the C*-normed algebra A0 [ · 0 ], but it is not necessarily a locally convex quasi *-algebra over the C*-algebra 0 [ · 0 ] (completion of A0 [ · 0 ]), since A 0 [ · 0 ] is not a locally convex A *-algebra under the topology τ . Furthermore, consider the conditions: (T3 ) For all λ ∈ , there exists λ ∈ , such that pλ (xy) ≤ x0 pλ (y), for all x, y ∈ A0 , with xy = yx.   0 + ) := x ∈ A 0 + : x0 ≤ 1 is τ -closed. (T4 ) The set U (A 0 [τ ]+ ∩ A 0 [ · 0 ] = A 0 [ · 0 ]+ , (T5 ) A +

0 [ · 0 ] and A 0 [τ ]+ 0 denotes the positive cone of the C*-algebra A where A 0 [τ ] (see the set of all positive elements of the locally convex quasi *-algebra A Definition 7.1.1, in Chap. 7 and discussion at the beginning of Sect. 6.2, from where 0 + . Note that 0 [τ ]+ is the τ -closure of A the respective notation is adopted); i.e., A 0 [τ ]+ is a wedge and not necessarily a positive cone. A Let A0 [ · 0 ] be a unital C*-normed algebra, τ a locally convex topology on A0 satisfying the conditions (T1 ) − (T5 ). Then, a quasi *-subalgebra A of the locally 0 [τ ], A0 ), containing A 0 [ · 0 ], is said to be a locally convex quasi *-algebra (A convex quasi C*-normed algebra over A0 and a locally convex quasi C*-algebra if A0 [ · 0 ] is a C*-algebra. The latter case will be discussed in detail in Chap. 7.

6.1 Representable Functionals on Locally Convex Quasi *-Algebras

145

 If π : A → L† (Dπ , Hπ ) is a *-representation of (A, A0 ), which is contin-

uous from A[τ ] into L† (Dπ , Hπ )[t], where t is a locally convex topology on L† (Dπ , Hπ ), we write for short, that π is (τ, t)-continuous.

In this regard, the GNS representation (see Sect. 2.4) will play a fundamental role in our analysis. When (A[τ ], A0 ) is a locally convex quasi *-algebra, taking into account the notation in Definition 2.4.6 (see also beginning of Sect. 3.2), we recall that   Rc (A, A0 ) := ω ∈ R(A, A0 ) : ω is continuous . We say that a representable linear functional ω on A (i.e., ω ∈ R(A, A0 )) is continuous if there exists a continuous seminorm p on A, such that |ω(a)| ≤ p(a),

∀ a ∈ A.

(6.1.1)

The notions of closable or closed form, given in Definition 3.1.46 in the Banach case, extend to the present situation too. Let ϕ be a positive sesquilinear form defined on A0 ×A0 . We say that ϕ is closable if for a net {xδ }δ∈ in A0 , one has xδ → 0 and ϕ(xδ − xγ , xδ − xγ ) → 0 ⇒ ϕ(xδ , xδ ) → 0. τ

Then, |ϕ(xδ , xδ )1/2 − ϕ(xγ , xγ )1/2 | ≤ ϕ(xδ − xγ , xδ − xγ )1/2 → 0, therefore {ϕ(xδ , xδ )}δ∈ is a Cauchy net. Thus, if ϕ is closable, it can be extended to a positive sesquilinear form ϕ defined on D(ϕ) × D(ϕ) by ϕ(a, a) := lim ϕ(xδ , xδ ), δ

where  D(ϕ) = a ∈ A : ∃ {xδ } ⊂ A0 with xδ → a and ϕ(xδ − xγ , xδ − xγ ) → 0 . τ

This definition extends in obvious way to pairs (a, b) with a, b ∈ D(ϕ). If ω is a positive linear functional on A0 , then we can define a positive sesquilinear form ϕω on A0 × A0 by ϕω (x, y) := ω(y ∗ x), x, y ∈ A0 . Now we prove the following Proposition 6.1.4 Let ω ∈ Rc (A, A0 ). Then, ϕω is closable. Proof Let xδ → 0 with ϕω (xδ − xγ , xδ − xγ ) → 0. Then, y ∗ xδ → 0, for every τ

τ

y ∈ A0 , since the multiplication is continuous (see Definition 6.1.1). The continuity

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of ω then, implies that ω(y ∗ xδ ) → 0, y ∈ A0 . Put   Nω := x ∈ A0 : ω(x ∗ x) = 0 . Then, A0 /Nω is a pre-Hilbert space under the well-defined inner product λω (x)|λω (y) := ω(y ∗ x),

x, y ∈ A0 ,

where λω (z) := z + Nω , z ∈ A0 . Let Hω denote the Hilbert space completion of A0 /Nω . The net {λω (xδ )} is Cauchy, since λω (xδ ) − λω (xγ )2 = ϕω (xδ − xγ , xδ − xγ ) → 0. Hence, it converges to some ξ ∈ Hω and λω (xδ )|λω (y) → ξ |λω (y),

∀ y ∈ A0 .

Moreover, λω (xδ )|λω (y) = ω(y ∗ xδ ) → 0,

∀ y ∈ A0 .

Thus, ξ |λω (y) = 0, for every y ∈ A0 . This implies that ξ = 0. Therefore,   ϕω (xδ , xδ ) → 0. Note that representability of ω is not used in the proof of Proposition 6.1.4. Consider now the set  D(ϕ ω ). AR := ω∈Rc (A,A0 )

If Rc (A, A0 ) = {0}, we put AR = A. Note that, if for every ω ∈ Rc (A, A0 ), ϕω is jointly continuous with respect to τ , we get AR = A. Proposition 6.1.5 AR is a vector subspace of A and A0 ⊂ AR . Moreover, if a ∈ AR and x ∈ A0 , then xa ∈ AR . Hence, if AR is *-invariant, then (AR , A0 ) is a quasi *-algebra. Proof We show that a ∈ AR and x ∈ A0 imply xa ∈ AR . Indeed, if ω ∈ Rc (A, A0 ), then also ωx ∈ Rc (A, A0 ), by (3.2.19). This implies that a ∈ D(ϕ ωx ) or, equivalently, xa ∈ D(ϕ ω ). Since ω is arbitrary, the statement is proved.  

6.2 Order Structure

147

6.2 Order Structure Given a locally convex quasi *-algebra (A[τ ], A0 ), we recall the concept of a positive element of A (see e.g., [23, pp. 21, 22]). This notion defines an order on the set Ah of all selfadjoint elements of A. The condition A+ ∩ (−A+ ) = {0}, on the positive elements A+ of A, implies that every non-zero element in A+ gives rise to a non-trivial continuous positive linear functional on A[τ ] (Theorem 6.2.2). The preceding condition is characterized in Proposition 6.2.7 and it itself, together with another condition that forces an element of A to be positive, show that (A[τ ], A0 ) attains enough (τ, tw )-continuous *-representations to separate its points (Corollary 6.2.10). Coming back to the given locally convex quasi *-algebra (A[τ ], A0 ), set (see also discussion after Corollary 3.1.24) +

A0 :=

 n 

 xk∗ xk ,

xk ∈ A0 , n ∈ N .

k=1 + Then, A+ 0 is a wedge in A0 and we call the elements of A0 positive elements of A0 . τ We call positive elements of A[τ ] the elements of A+ and we denote them by A+ . 0 +τ + That is, A := A0 . The set A+ is a qm-admissible wedge (generalization of m-admissible wedge given by Schmüdgen [23, p. 22]), in the following sense:

1. 2. 3. 4.

e ∈ A+ , if (A[τ ], A0 ) has a unit e; a + b ∈ A+ , ∀ a, b ∈ A+ ; λa ∈ A+ , ∀ a ∈ A+ , λ ≥ 0; x ∗ ax ∈ A+ , ∀ a ∈ A+ , x ∈ A0 .

  Clearly, A+ defines an order on the real vector space Ah = x ∈ A : x = x ∗

by x ≤ y ⇔ y − x ∈ A+ . For a ∈ A+ , we shall often use the notation a ≥ 0. The following proposition is straightforward. Proposition 6.2.1 If a ≥ 0, then π(a) ≥ 0, for every (τ, tw )-continuous *representation of (A[τ ], A0 ). The theorem that follows, shows that if the set A+ is proper, then (A[τ ], A0 ) attains non-trivial continuous positive linear functionals, in the sense of Definition 7.1.3, in Chap. 7.

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Theorem 6.2.2 Assume that A+ ∩ (−A+ ) = {0}. Let a ∈ A+ , a = 0. Then, there exists a continuous linear functional ω on A[τ ] with the properties: (i) ω(b) ≥ 0, ∀ b ∈ A+ ; (ii) ω(a) > 0. Proof Consider the real vector space Ah and a ∈ A+ \ {0}. The set {a} is obviously convex and compact and does not intersect (−A+ ). Hence by [5, Ch.2, §5, Proposition 4], there exists a closed hyperplane separating these two sets. Let g(x) = 0 be the equation of this hyperplane. Then, either g(a) > 0 with g(−A+ ) < 0 (in which case we take ω = g), or the contrary (where, in this case, we take ω = −g) .   Theorem 6.2.2 leads to the following (see also discussion before Lemma 3.1.48) Definition 6.2.3 A linear functional ω on A is called positive if ω(a) ≥ 0, ∀ a ∈ A+ . The next Proposition 6.2.4, provides conditions under which continuous linear functionals on (A[τ ], A0 ) are positive and hermitian. Proposition 6.2.4 Assume that AR = A and that (A[τ ], A0 ) has a unit. Then, every continuous linear functional ω on A, such that ω(x ∗ x) ≥ 0, for every x ∈ A0 , is positive and hermitian. Proof Since ω is positive on A0 , it is hermitian on A0 . Thus, by continuity of ω and continuity of the involution (see Definition 6.1.1) we are done.   Using representable linear functionals, the following proposition provides invariant positive sesquilinear forms with fixed core, i.e., elements of the set IA0 (A) (see, Proposition 2.3.2 and Remark 2.3.3). Proposition 6.2.5 Assume that AR = A. Then, for every ω ∈ Rc (A, A0 ), ϕ ω ∈ IA0 (A); i.e., ϕ ω is an ips-form on A with core A0 . Proof Let N(ϕ ω ) = {a ∈ A : ϕ ω (a, a) = 0} and Hϕ ω the Hilbert space obtained by completing A/N (ϕ ω ) with respect to the (well-defined) inner product λϕ ω (a)|λϕ ω (b) := ϕ ω (a, b), a, b ∈ A, where λϕ ω (a) := a + N (ϕ ω ). According to the discussion before Remark 2.3.3 and definition of ϕ ω ’s, we need to show that λϕ ω (A0 ) is dense in Hϕ ω . Assume, on the contrary, that there exists a ∈ A, such that λϕ ω (x)|λϕ ω (a) = 0, for every x ∈ A0 .  τ Since a ∈ ω∈Rc (A,A0 ) D(ϕ ω ) = A, there exists a net {xδ } ⊂ A0 , such that xδ → a and ϕ ω (xδ − a, xδ − a) → 0. Hence, ϕ ω (a, a) = lim ϕ ω (xδ , xδ ) = 0. δ

6.2 Order Structure

149

Consequently λϕ ω (a) = 0. The proof is now complete according to Proposi  tion 2.3.2. Definition 6.2.6 A family of positive linear functionals F on (A[τ ], A0 ) is called sufficient if for every a ∈ A+ , a = 0, there exists ω ∈ F, such that ω(a) > 0. Proposition 6.2.7 Let (A[τ ], A0 ) be a locally convex quasi *-algebra. The following statements are equivalent: (i) A+ ∩ (−A+ ) = {0}; (ii) Rc (A, A0 ) is sufficient. Proof (i) ⇒ (ii) This is Theorem 6.2.2. (ii) ⇒ (i) Let a ∈ A+ ∩ (−A+ ) and ω ∈ Rc (A, A0 ). Then, ω is a continuous positive linear functional on A, therefore ω(a) ≥ 0 and ω(−a) = −ω(a) ≥ 0. Thus ω(a) = 0. Since ω is arbitrary, we finally get a = 0.   It is clear from Theorem 6.2.2 and Definition 6.2.6 that if A+ ∩ (−A+ ) = {0}, then the family of all continuous positive linear functionals on (A[τ ], A0 ) is sufficient. Proposition 6.2.8 Let (A[τ ], A0 ) be a locally convex quasi *-algebra with Rc (A, A0 ) sufficient. Assume that the following condition (P) holds: (P) b ∈ A and ω(x ∗ bx) ≥ 0, f or all ω ∈ Rc (A, A0 ) and x ∈ A0 imply b ∈ A+ . Then, for an element a ∈ A, the following statements are equivalent: (i) a ∈ A+ ; (ii) ω(a) ≥ 0, for every ω ∈ Rc (A, A0 ); (iii) π(a) ≥ 0, for every (τ, tw )-continuous *-representation π of (A[τ ], A0 ). Proof (i) ⇒ (ii) is an easy consequence of the definition of positive elements and the continuity of the elements of Rc (A, A0 ) with respect to τ . (ii) ⇒ (iii) Let π be a (τ, tw )-continuous *-representation of (A[τ ], A0 ). Define ωξ (a) := π(a)ξ |ξ , a ∈ A, with ξ ∈ D, ξ  = 1. Then, ωξ ∈ Rc (A, A0 ), since |ωξ (a)| = |π(a)ξ |ξ | ≤ p(a), ∀ a ∈ A, for some τ -continuous seminorm p on A. Thus, if a satisfies (ii), π(a)ξ |ξ  ≥ 0, for every ξ ∈ D, which proves (iii). (iii) ⇒ (i) Let ω ∈ Rc (A, A0 ) and let πω be the corresponding GNS representation. Then, πω is (τ, tw )-continuous. Indeed, due to the continuity of ω and (ii) of Definition 6.1.1, we get |πω (a)λω (x)|λω (y)| = |ω(y ∗ ax)| ≤ p(a),

∀ a ∈ A, x, y ∈ A0 ,

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for some τ -continuous seminorm p on A. Applying (iii) we have πω (a) ≥ 0. This implies that ω(x ∗ ax) ≥ 0, for every x ∈ A0 . The statement now follows from the assumption (P).   Remark 6.2.9 1. If (A[τ ], A0 ) has a unit, then the equivalence of (ii) and (iii) does not depend on (P). In this case, (P) is equivalent to the following (P ) If b ∈ A and ω(b) ≥ 0, for every ω ∈ Rc (A, A0 ), then b ∈ A+ . Indeed, since we have unit (P) implies (P ). On the other hand, by (3.2.19), for any ω ∈ Rc (A, A0 ) and x ∈ A0 we have that ωx ∈ Rc (A, A0 ), where ωx (b) := ω(x ∗ bx), b ∈ A, so that (P ) implies (P). 2. The condition (P) together with A+ ∩ (−A+ ) = {0} implies that, for every 0 = a ∈ A, there exists ω ∈ Rc (A, A0 ), such that ω(a) = 0. Indeed, if ω(a) = 0, for every ω ∈ Rc (A, A0 ), then (Proposition 6.2.8) a ∈ A+ and −a ∈ A+ ; hence a = 0, a contradiction. From Remark 6.2.9(2) and the proof of Proposition 6.2.8 we now have the following Corollary 6.2.10 Let (A[τ ], A0 ) be a locally convex quasi *-algebra with unit e. Suppose that Rc (A, A0 ) is sufficient and that condition (P) of Proposition 6.2.8 is fulfilled. Then, for every 0 = a ∈ A, there is a (τ, tw )-continuous *-representation π of (A[τ ], A0 ), namely π = πω , ω ∈ Rc (A, A0 ), such that π(a) = 0.

6.3 Fully Representable Quasi *-Algebras In Sect. 6.2, given a locally convex quasi *-algebra (A[τ ], A0 ), we have seen that the sufficiency of the set Rc (A, A0 ) together with the condition AR = A equip the given algebra with important properties (cf., for instance, Theorem 6.2.2, Proposition 6.2.4 and Corollary 6.2.10) that are very close to the properties that C*-algebras enjoy and offer to them their rich structure. All these lead us to the following Definition 6.3.1 A locally convex quasi *-algebra (A[τ ], A0 ) is called fully representable if Rc (A, A0 ) is sufficient and AR = A. Example 6.3.2 Let I = [0, 1]. Consider the Banach quasi *-algebra (Lp (I ), L∞ (I )). Then, every ω ∈ Rc (Lp (I ), L∞ (I )) has the form 

1

ω(f ) =

f (t)v(t)dt, 0

f ∈ Lp (I ),

6.3 Fully Representable Quasi *-Algebras

151

with v ∈ Lp/p−2 (I ), v ≥ 0 and p ≥ 2. One readily checks that ω satisfies the conditions (L1) and (L2) of Definition 2.4.6. It is easily seen that the condition (L3) of the same definition is also fulfilled. Conversely, assume that (L3) is satisfied; i.e., for every f ∈ Lp (I ) there exists γ > 0, such that   |ω(f α)| = 

1

∗

0

  f (t)α(t)v(t)dt  ≤ γ



1

!1/2 |α(t)| v(t)dt 2

,

∀ α ∈ L∞ (I ).

0

This implies that f ∈ L2 (I, vdt) and γ = f 2,v . Hence, in order that (L3) be satisfied for every f ∈ Lp (I ), we must have v ∈ Lp/p−2 (I ). Hence, if p ≥ 1, ω is representable, if and only if, v ∈ Lp/p−2 (I ). If v ∈ Lp/p−1 (I ) \ Lp/p−2 (I ), then ω is continuous but not representable. If 1 ≤ p < 2, the condition Lp (I ) ⊂ L2 (I, vdt) is not satisfied for every non zero v. In this case, there are no continuous representable functionals. Let us now come back to the case p ≥ 2. Let v ∈ Lp/p−2 (I ). We want to determine ϕ ω , where 

1

ϕω (α, β) =

α(t)β(t)v(t)dt,

α, β ∈ L∞ (I ).

0

Let f ∈ D(ϕ ω ). Then, there exists a sequence {αn } ⊂ L∞ (I ), such that p



1

αn → f and

|αn (t) − αm (t)|2 v(t)dt → 0.

0

Hence, there exists v0 ∈ L2 (I, vdt), such that αn → v0 , in the L2 (I, vdt)-norm. This, in turn, implies that f = v0 , almost everywhere. Thus, D(ϕ ω ) = Lp (I ) ∩ L2 (I, vdt). Therefore, Lp (I )R =

 ω∈Rc

(Lp ,L∞ )

⎛ D(ϕ ω ) = Lp (I ) ∩ ⎝



⎞ L2 (I, vdt)⎠ .

v∈Lp/p−2 (I )

But f ∈ L2 (I, vdt), for every v ∈ Lp/p−2 (I ), if and only if, f ∈ Lp (I ). In conclusion, for p ≥ 2, Lp (I )R = Lp (I ) and (Lp (I ), L∞ (I )) is fully representable. The example above is a particular case of the next Theorem 6.3.3. Since, in the familiar case discussed therein, everything can be easily computed, we have preferred an explicit proof there. It is implicitly proved in this example that (Lp (I ), L∞ (I )) has sufficiently many continuous representable linear functionals. Moreover, Theorem 6.3.3 also shows that in the case of Banach quasi *-algebras there is close relationship between full representability and *-semisimplicity.

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Theorem 6.3.3 Let (A[ · ], A0 ) be a Banach quasi *-algebra with unit e. The following statements are equivalent: (i) Rc (A, A0 ) is sufficient; (ii) (A, A0 ) is fully representable. If the condition (P) holds, (i) and (ii) are equivalent to the following (iii) (A[ · ], A0 ) is *-semisimple. Proof The equivalence between (i) and (ii) follows from the very definitions and from Proposition 3.2.2. Suppose now that the condition (P ) holds. We shall prove that (iii) is equivalent to (ii) ∼ (i). (ii) ⇒ (iii) First, we notice that every ϕ ∈ SA0 (A) can be written as ϕ ω , for some ω ∈ Rc (A, A0 ). Indeed, if we put ωϕ (a) := ϕ(a, e),

a ∈ A,

then it is easily seen that ωϕ is continuous and representable, so ωϕ ∈ Rc (A, A0 ) and ϕ ωϕ = ϕ. On the other hand, consider a linear functional 0 = ω ∈ Rc (A, A0 ) and let ϕ ω be the sesquilinear form associated to it as in (3.2.16). By Proposition 3.2.2 D(ϕ ω ) = A, thus ϕ ω is bounded. If we put ϕω = ϕ ω /ϕ ω , then ϕω ∈ SA0 (A). Let a ∈ A be such that ϕ(a, a) = 0, for every ϕ ∈ SA0 (A). For what we have just shown, it is enough to prove that, if ϕ ω (a, a) = 0, for every ω ∈ Rc (A, A0 ), then a = 0. We have |ω(a)| = |ϕ ω (a, e)| ≤ ϕ ω (e, e)1/2 ϕ ω (a, a)1/2 = 0. Thus ω(a) = 0, for every ω ∈ Rc (A, A0 ). By condition (P), a ≥ 0; but since Rc (A, A0 ) is sufficient, we get a = 0. (iii) ⇒ (i) Let a ∈ A+ and suppose that ω(a) = 0, for every ω ∈ Rc (A, A0 ). If x ∈ A0 , then the linear functional ωx defined by ωx (c) := ω(x ∗ cx), c ∈ A, is representable and continuous. Thus, ωx (a) = 0, for every x ∈ A0 . By polarization, one can easily conclude that ω(y ∗ ax) = 0, for every x, y ∈ A0 . This implies that ϕ ω (ax, y) = 0, for every x, y ∈ A0 . Now choose x = e and take a sequence {yn } ⊂ A+ 0 converging to a. Then ϕ ω (a, a) = limn→∞ ϕ ω (a, yn ) = 0. As in the proof of (ii) ⇒ (iii) we conclude that ϕ(a, a) = 0, for every ϕ ∈ SA0 (A). Hence   a = 0. Example 6.3.4 The space S  (R) of tempered distributions may be regarded as a locally convex quasi *-algebra over the *-algebra S(R). S  (R) is the dual of S(R), when the latter is endowed with the locally convex topology t defined by the family of seminorms pk,r (f ) = sup |x k D r f (x)|, f ∈ S(R), k, r ∈ N. x∈R

6.3 Fully Representable Quasi *-Algebras

153

The (partial) multiplication in S  (R) is defined by (F · f )(g) = (f · F )(g) = F (f g), F ∈ S  (R), f, g ∈ S(R). The space S  (R) is endowed with the strong dual topology t . Since S  (R)[t ] is reflexive, every continuous functional ω on S  (R)[t ] has the form ω(F ) = ωf (F ) := F (f ), for some f ∈ S(R). Also in this case there are no nontrivial continuous representable functionals on S  (R). Indeed, (L3) of Definition 2.4.6 is never satisfied by nonzero positive functionals ωf , f ≥ 0, since, if for every F ∈ S  (R), there exists γF > 0, such that ωf (F ∗ · g) ≤ γF ωf (g ∗ g)1/2 ,

∀ g ∈ S(R),

then 

|F ∗ (gf )| ≤ γF

R

g ∗ (x)g(x)f (x)dx

!1/2 = γF g2,f ,

where  · 2,f denotes the norm of L2 (R, f dx). This implies that there exists h ∈ L2 (R, f dx), such that F ∗ (gf ) =

 R

h(x)g(x)f (x)dx,

∀ g ∈ S(R).

Hence, F ∗ restricted to the linear subspace {gf : g ∈ S(R)} acts as a function. This is a contradiction if f (and then ωf ) is nonzero. Example 6.3.5 (Quasi *-Algebras of Operators) Let (L† (D, H), L† (D)b ) be the locally convex quasi *-algebra of Example 2.1.3. Let ξ ∈ D. Then, the positive linear functional ωξ (A) = Aξ |ξ , A ∈ L† (D, H) is representable. The corresponding sesquilinear form ϕωξ on L† (D)b × L† (D)b is jointly continuous with respect to the topology τs∗ , so that D(ϕωξ ) = L† (D, H). The same is true in the more general case, where ω(A) =

n  Aξi |ξi , ξi ∈ D, i = 1, . . . , n. i=1

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6 Locally Convex Quasi *-Algebras

Let us now consider (L† (D, H), L† (D)b ), where L† (D, H) is endowed with the weak topology tw . Then, the following statements hold [39, 83]: (i) Every weakly continuous (or strongly∗ -continuous) linear functional ω on L† (D, H) has the form ω(A) =

n  Aξi |ηi , ξi , ηi ∈ D, i = 1, . . . , n.

(6.3.2)

i=1

(ii) Every weakly continuous positive linear functional ω on L† (D, H) has the form n  Aζi |ζi , ζi ∈ D, i = 1, . . . , n. ω(A) =

(6.3.3)

i=1

Both locally convex quasi *-algebras (L† (D, H)[ts ∗ ], L† (D)b ) and (L† (D, H) [tw ], L† (D)b ) satisfy the equality L† (D, H)R = L† (D, H) and therefore both (L† (D, H)[ts ∗ ], L† (D)b ) and (L† (D, H)[tw ], L† (D)b ) are fully representable. Indeed, let ω be ts ∗ -continuous and representable. If A ∈ L† (D, H), there exists a net {Aδ } of elements of L† (D)b , such that Aδ → A. Then, using the representation ts ∗

(6.3.3) we have, ϕω (Aδ − Aγ , Aδ − Aγ ) = ω((Aδ − Aγ )∗ (Aδ − Aγ )) n  (Aδ − Aγ )ζi |(Aδ − Aγ )ζi  = i=1

=

n 

(Aδ − Aγ )ζi 2 → 0.

i=1

This proves that every A ∈ L† (D, H) is in the domain of the closure of ϕω , with respect to the topology ts ∗ . The statement for the weak topology follows by observing that if ω is weakly continuous, then it is automatically ts ∗ -continuous. Note that in the Examples 6.3.2 and 6.3.5 the condition (P) is satisfied, while for the Example 6.3.4 it is meaningless. We do not know whether (P) always holds, when Rc (A, A0 ) is sufficient.

6.4 Ordered Bounded Elements The concept of a bounded element in a locally convex algebra was first introduced by G.R. Allan [32], (1965), for building a spectral theory for this kind of algebras. Similar definitions had been considered earlier by S. Warner [94], in the case of m-convex algebras, as well as by L. Waelbroeck [93] in a specific framework and

6.4 Ordered Bounded Elements

155

under the assumption of quasi-completeness and commutativity of the topological algebras involved. Bounded elements in the context of quasi *-algebras have been already considered in Sect. 4.1. For an extension to partial *-algebras, see [39]. K. Schmüdgen has considered Allan’s bounded elements in his research on the unbounded operator algebras called O*-algebras and recently (2005), the same author considered bounded elements in a purely algebraic sense (see also [92]) and studied the structure of the set of the introduced bounded elements, in order to use them for proving a “strict Positivstellensatz for the Weyl algebra” [75]. Motivated from this, and having in hands the results of Sect. 6.2, we introduce the concept of “order boundedness” and what is interesting, is that this concept coincides under some conditions with the usual notion of boundedness, which one gets when the *-algebra under consideration admits *-representations (see, for instance, Proposition 6.4.4, Theorem 6.4.5 and Corollary 6.4.8). Furthermore, for suitable fully representable locally convex quasi *-algebras (A[τ ], A0 ), considering or the set Aor b of all order bounded elements of A[τ ], we prove that Ab becomes either a partial C*-algebra or a C*-algebra (Theorem 6.4.16). Let (A[τ ], A0 ) be an arbitrary locally convex quasi *-algebra. As we have seen in Sect. 6.2, (A[τ ], A0 ) has a natural order related to the topology τ . This order can be used to define bounded elements. In what follows, we shall assume that (A, A0 ) has a unit e. Let a ∈ A; put ((a) = 12 (a + a ∗ ), )(a) = 2i1 (a − a ∗ ). Then, ((a), )(a) ∈ Ah and a = ((a) + i)(a). Definition 6.4.1 An element a ∈ A is called order bounded if there exists γ ≥ 0, such that ±((a) ≤ γ e,

±)(a) ≤ γ e.

We denote by Aor b the set of all order bounded elements of A[τ ]. In this regard, we have Proposition 6.4.2 The following statements hold: (1) (2) (3) (4)

αa + βb ∈ Aor ∀ a, b ∈ Aor b , b , α, β ∈ C; or ∗ a ∈ Ab ⇔ a ∈ Aor b ; or or a ∈ Aor b , x ∈ Ab ∩ A0 ⇒ xa ∈ Ab ; or ∗ or x ∈ Ab ∩ A0 ⇔ xx ∈ Ab ∩ A0 .

or Hence, (Aor b , Ab ∩ A0 ) is a quasi *-algebra.

Proof The proof is similar to that of [75, Lemma 2.1]. For a ∈

(Aor ) b

h,

put   aor b := inf γ > 0 : −γ e ≤ a ≤ γ e .

or Then,  · or b is a seminorm on the real vector space (Ab )h .

 

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6 Locally Convex Quasi *-Algebras

or Lemma 6.4.3 If A+ ∩ (−A+ ) = {0},  · or b is a norm on (Ab )h .

Proof Put E = {γ > 0 : −γ e ≤ a ≤ γ e}. If inf E = 0, then for every ε > 0, there exists γε ∈ E, such that γε < ε. This implies that −εe ≤ a ≤ εe. If ω ∈ Rc (A, A0 ), we get −ε ω(e) ≤ ω(a) ≤ ε ω(e) (we may suppose ω(e) > 0, for every ω ∈ Rc (A, A0 ), since from (L3) of Definition 2.4.6, if ω(e) = 0, then ω ≡ 0). Hence, ω(a) = 0. By the sufficiency of Rc (A, A0 ) (Definition 6.2.6), it follows   that a = 0. Proposition 6.4.4 If a ∈ Aor b , then π(a) is a bounded operator, for every (τ, tw )continuous *-representation π of (A[τ ], A0 ). Moreover, if a = a ∗ , then π(a) ≤ aor b . Proof It follows easily from Proposition 6.2.1 and the very definitions.

 

Theorem 6.4.5 Let (A[τ ], A0 ) be fully representable and assume that condition (P) holds. Then, for a ∈ A, the following statements are equivalent: (i) a is order bounded; (ii) there exists γa > 0, such that |ω(x ∗ ax)| ≤ γa ω(x ∗ x),

∀ ω ∈ Rc (A, A0 ), x ∈ A0 ;

(iii) there exists γa > 0, such that |ω(y ∗ ax)| ≤ γa ω(x ∗ x)1/2 ω(y ∗ y)1/2 ,

∀ ω ∈ Rc (A, A0 ), x, y ∈ A0 .

Proof It is sufficient to consider the case a = a ∗ . Also, as in the proof of Lemma 6.4.3, we suppose ω(e) > 0, for every ω ∈ Rc (A, A0 ). (i) ⇒ (ii) If a = a ∗ is bounded, there exists γ > 0, such that −γ e ≤ a ≤ γ e. Hence, from Proposition 6.2.8, for every ω ∈ Rc (A, A0 ), ω(γ e − a) ≥ 0. It follows that ω(x ∗ (γ e − a)x) ≥ 0. Thus, ω(x ∗ ax) ≤ γ ω(x ∗ x), for every x ∈ A0 . Similarly we can show that −γ ω(x ∗ x) ≤ ω(a ∗ xa), for every x ∈ A0 . (ii) ⇒ (i) Assume now that there exists γa > 0 such that |ω(x ∗ ax)| ≤ γa ω(x ∗ x),

∀ ω ∈ Rc (A, A0 ), x ∈ A0 .

Define γ := sup{|ω(x ∗ ax)| : ω ∈ Rc (A, A0 ), x ∈ A0 with ω(x ∗ x) = 1}.  Then (see Proposition 6.2.8 and Remark 6.2.9(1)), for an arbitrary ω ∈ Rc (A, A0 ), we get, ω ( γ e ± a) =  γ ω (e) ± ω (a) = ω (e)( γ ± ω (u∗ au)) ≥ 0, where u =

e . ω (e)1/2

6.4 Ordered Bounded Elements

157

Hence, ω ( γ e ± a) ≥ 0, for every ω ∈ Rc (A, A0 ). Then, by Remark 6.2.9, − γe ≤ a ≤  γ e; i.e., a is order bounded. (i) ⇒ (iii) The GNS representation πω is (τ, tw )-continuous, hence if a = a ∗ ∈ A, by Proposition 6.4.4, πω (a) is bounded. Thus, |ω(y ∗ ax)| = |πω (a)λω (x)|λω (y)| ≤ πω (a) λω (x) λω (y) ∗ 1/2 ≤ aor ω(y ∗ y)1/2 , b ω(x x)

∀ ω ∈ Rc (A, A0 ), x, y ∈ A0 .  

(iii) ⇒ (ii) is obvious. Remark 6.4.6 The proof above shows that for a = a ∗ ,   ∗ ∗ aor b ≤ sup |ω(x ax)| : ω ∈ Rc (A, A0 ), x ∈ A0 with ω(x x) = 1 .

Corollary 6.4.7 Let (A[τ ], A0 ) be fully representable. If a is order bounded, there exists γa > 0, such that |ϕ ω (ax, c)| ≤ γa ω(x ∗ x)1/2 ϕ ω (c, c)1/2 ,

∀ ω ∈ Rc (A, A0 ), x ∈ A0 , c ∈ A.

 Proof Let a ∈ A be order bounded. Since ω∈Rc (A,A0 ) D(ϕ ω ) = A, for every c ∈ A, there exists a net {zδ } ⊂ A0 , such that zδ → c and ϕ ω (c − zδ , c − zδ ) → 0. τ Then, by Theorem 6.4.5(iii), we get |ϕ ω (ax, c)| = lim |ϕ ω (xa, zδ )| ≤ γa ω(x ∗ x)1/2 lim ϕ ω (zδ , zδ )1/2 δ

δ

∗

= γa ω(x x)

1/2

ϕ ω (c, c)1/2 ,

∀ x ∈ A0 .  

Corollary 6.4.8 Let (A[τ ], A0 ) be fully representable and a ∈ A. Then, a is order bounded, if and only if, there exists γa > 0 such that ϕω (ax, ax) ≤ γa2 ω(x ∗ x),

∀ ω ∈ Rc (A, A0 ), x ∈ A0 .

Proof The necessity follows by putting c = ax in the inequality of Corollary 6.4.7. The sufficiency is clear.   Let a be order bounded. Define  q(a) := sup |ω(y ∗ ax)| : ω ∈ Rc (A, A0 ), x, y ∈ A0 with ω(x ∗ x) = ω(y ∗ y) = 1 . Then, we have ∗ or Lemma 6.4.9 q(a) = aor b , for every a = a ∈ Ab .

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6 Locally Convex Quasi *-Algebras

Proof The inequality aor b ≤ q(a) follows from Remark 6.4.6. Let γ > 0, such that −γ e ≤ a ≤ γ e. Then, by the proof of Theorem 6.4.5, we have, q(a) ≤ γ ;   whence the statement follows. Lemma 6.4.9 shows that q extends  · or b . For this reason, we will use the symbol or or  · or b for q too. It is easily seen that  · b is a norm on Ab . An easy consequence of the above statements is now the following Proposition 6.4.10 For every a ∈ Aor b ,  ∗ = sup ϕ (ax, ax) : ω ∈ R (A, A ), x ∈ A with ω(x x) = 1 . aor ω c 0 0 b

6.4.1 Partial Multiplication If (A[τ ], A0 ) is fully representable we can introduce on A a partial multiplication, which makes it into a partial *-algebra. This multiplication extends that one introduced before Proposition 3.1.31 for *-semisimple Banach quasi *-algebras. For convenience, we keep the same symbol, , as there. Definition 6.4.11 Let (A[τ ], A0 ) be fully representable. The weak product a b of two elements a, b ∈ A is well defined if there exists c ∈ A such that ϕ ω (bx, a ∗ y) = ϕ ω (cx, y),

∀ ω ∈ Rc (A, A0 ), x, y ∈ A0 .

In this case, we put a b := c. We call  weak multiplication on A. Since Rc (A, A0 ) is sufficient, the element c is unique. Proposition 6.4.12 (A[τ ], A0 ) endowed with the weak multiplication  is a partial *-algebra with A0 ⊂ RA. Proposition 6.4.13 Let a, b be order bounded elements of A. The following statements hold: or (i) a ∗ is order bounded too, and a ∗ or b = ab ; (ii) If a b is well-defined, then a b is order bounded and or or a bor b ≤ ab bb .

Proof (i) The first part of (i) is given by Proposition 6.4.2(2). The second part follows from the property (L2) (Definition 2.4.6) of an ω ∈ Rc (A, A0 ), from Corollary 6.4.8 and the definition of  · or b .

6.4 Ordered Bounded Elements

159

(ii) If a b, a, b ∈ A, is well-defined, then for every ω ∈ Rc (A, A0 ), Corollary 6.4.7 implies |ϕ ω ((a b)x, y)| = |ϕ ω (bx, a ∗ y)| ≤ ϕ ω (bx, bx)1/2 ϕ ω (a ∗ y, a ∗ y)1/2 or 1/2 ≤ aor ϕ ω (y, y)1/2 , b bb ϕ ω (x, x)

∀ x, y ∈ A0 .

Taking now sup on the left hand side (see Proposition 6.4.10), we get the desired inequality.   We recall that an unbounded C*-seminorm p on a partial *-algebra A is a seminorm defined on a partial *-subalgebra D(p) of A, the domain of p, with the properties: • p(ab) ≤ p(a)p(b), whenever ab is well-defined; • p(a ∗ a) = p(a)2 , whenever a ∗ a is well-defined (see, e.g., [2, 46, 79]). Proposition 6.4.14  · or b is an unbounded C*-norm on A with domain Ab . Proof This can be deduced from [87, Proposition 2.6].

 

It is worth mentioning here that certain unbounded C*-seminorms give rise to “wellbehaved” (unbounded) *-representations (for more details, see [2, Chapter 8] and [46, 79]). Now having (A[τ ], A0 ) to be fully representable, we can endow A with the strong and strong∗ topology, where both are defined in a natural way through the elements of Rc (A, A0 ). Indeed: • the strong topology τs , is defined by the family of seminorms a ∈ A → ϕ ω (a, a)1/2 , ω ∈ Rc (A, A0 ), • the strong∗ topology τs ∗ , is respectively defined by the family of seminorms  a ∈ A → max ϕ ω (a, a)1/2 , ϕ ω (a ∗ , a ∗ )1/2 , ω ∈ Rc (A, A0 ). Definition 6.4.15 Let A be a partial *-algebra. We say that A is a partial C*algebra if A is a Banach space under a norm  ·  satisfying the following properties: (i) a ∗  = a, ∀ a ∈ A; (ii) ab ≤ a b, whenever ab is well-defined; (iii) a ∗ a = a2 , whenever a ∗ a is well-defined.

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6 Locally Convex Quasi *-Algebras

The theorem that follows, shows that the quasi *-algebra (Ab , Ab ∩ A0 ) (see Proposition 6.4.2), under certain conditions achieves a very rich structure. Theorem 6.4.16 Let (A[τ ], A0 ) be a fully representable locally convex quasi *algebra with unit e. Assume that A is τs ∗ -complete. Then, Ab is a partial C*-algebra under the weak multiplication  and the norm  · or b . Assume, in addition, that (R) If a, b ∈ A and πω (a)πω (b) is well-defined, for every ω ∈ Rc (A, A0 ), then there exists c ∈ A, such that πω (a)πω (b) = πω (c), for every ω ∈ Rc (A, A0 ). Then, Ab is a C*-algebra with the weak multiplication  and the norm  · or b . Proof Since  · or b satisfies (i)–(iii) of Definition 6.4.15 on Ab (see e.g., Proposition 6.4.13), we need only to prove the completeness of Ab . ∗ Let {an } be a Cauchy sequence with respect to the norm  · or b . Then, {an } is or  · b -Cauchy too. Hence, for every ω ∈ Rc (A, A0 ) and x ∈ A0 we have

ϕ ω (an − am )x, (an − am )x → 0, as n, m → ∞ and ∗ )x, (a ∗ − a ∗ )x → 0, as n, m → ∞. ϕ ω (an∗ − am n m Therefore, {an } is also Cauchy with respect to τs ∗ . Then, since A is τs ∗ complete, there exists a ∈ A such that an → a. Moreover, τs ∗



2 ϕ ω (ax, ax) = lim ϕ ω (an x, an x) ≤ lim sup an or ϕ ω (x, x), ∀ x ∈ A0 , b n→∞

n→∞



2 < ∞ (by the boundedness of the sequence {an or with lim supn→∞ an or b b }), so by Corollary 6.4.8, we conclude that a is order bounded. Finally, by the Cauchy condition, for every ε > 0, there exists nε ∈ N, such that an − am or b < ε, for every n, m > nε . This implies that

ϕ ω (an − am )x, (an − am )x < ε ϕ ω (x, x),

∀ ω ∈ Rc (A, A0 ), x ∈ A0 .

Then, if we fix n > nε and let m → ∞, we obtain

ϕ ω (an − a)x, (an − a)x ≤ ε ϕ ω (x, x),

∀ ω ∈ Rc (A, A0 ), x ∈ A0 .

This, in turn, implies that an − aor b ≤ ε, for every n ≥ nε . So completeness of Ab [ · or ] is proved. b Now, assume that condition (R) holds. By Proposition 6.4.4 it follows that if a, b ∈ Ab , then the operators πω (a), πω (b) are bounded, therefore the operator πω (a)πω (b) is well-defined, hence (Proposition 6.4.13) bounded. Thus, by (R), there exists c ∈ A, such that πω (a)πω (b) = πω (c), for every ω ∈ Rc (A, A0 ).

6.4 Ordered Bounded Elements

161

Furthermore, for every ω ∈ Rc (A, A0 ) and x, y ∈ A0 , we have ϕ ω (bx, a ∗ y) = πω (b)λω (x)|πϕ (a ∗ )λω (y) = πω (a)πω (b)λω (x)|λω (y) = πω (c)λω (x)|λω (y) = ϕ ω (cx, y). Hence a b is well-defined (Definition 6.4.11). Thus (see also Proposition 6.4.14), Ab is a C*-algebra.   Example 6.4.17 Let us consider again the locally convex quasi *-algebra (L† (D, H), L† (D)b ) of example 6.3.5. As proved there, this quasi *-algebra is fully-representable and from (6.3.3) it follows that the topology τs ∗ defined before Definition 6.4.15 coincides with the strong∗ topology ts ∗ of L† (D, H). One can prove easily that an element T ∈ L† (D, H) is order bounded, if and only if, T ∈ B(H). So that,   (L† (D, H))b = T ∈ L† (D, H) : T is a bounded operator . This is clearly a C*-algebra as expected by Theorem 6.4.16.

Chapter 7

Locally Convex Quasi C*-Algebras and Their Structure

7.1 Locally Convex Quasi C*-Algebras Throughout this chapter A0 [·0 ] denotes a unital C*-algebra and τ a locally convex 0 [τ ] denote the completion of A0 with respect to the topology topology on A0 . Let A 0 [τ ], containing A0 , will form τ . Under certain conditions on τ , a subspace A of A (together with A0 ) a locally convex quasi *-algebra (A[τ ], A0 ), which is named locally convex quasi C*-algebra. Examples and basic properties of such algebras are presented. So, let A0 [ · 0 ] and τ be as before, with {pλ }λ∈ a defining family of seminorms for τ . Suppose that τ satisfies the properties: (T1 ) A0 [τ ] is a locally convex *-algebra with separately continuous multiplication. (T2 ) τ &  · 0 . Then, the identity map A0 [ · 0 ] → A0 [τ ] extends to a continuous *-linear map 0 [τ ] and by (T2 ), the C*-algebra A0 [ · 0 ] can be regarded embedded A0 [ · 0 ] → A 0 [τ ]. It is easily shown that A 0 [τ ] is a locally convex quasi *-algebra over A0 into A (cf. Definition 6.1.1 and [59, Section 3]). The next Definition 7.1.1 provides concepts of positivity for elements of a quasi 0 [τ ]. In this regard, see also Example 6.1.3 and beginning of Sect. 6.2. *-algebra A 0 [τ ] is called positive (resp. commutatively Definition 7.1.1 An element a of A positive) if there is a net (resp. commuting net) {xα }α∈ of the positive cone A+ 0 of the C*-algebra A0 [ · 0 ], which converges to a with respect to the topology τ . 0 [τ ], we shall The set of all positive (resp. commutatively positive) elements of A 0 [τ ]+ (resp. A 0 [τ ]+ denote by A ). c We have already used the symbol A+ 0 for the set of all positive elements of the C*-algebra A0 [ · 0 ]. It is worth noticing that the notion of a positive element of 0 [τ ] given here is exactly the same as that introduced in Sect. 6.2, since in the C*A algebra A0 [·0 ] the set of positive elements (i.e., elements with spectrum contained © Springer Nature Switzerland AG 2020 M. Fragoulopoulou, C. Trapani, Locally Convex Quasi *-Algebras and their Representations, Lecture Notes in Mathematics 2257, https://doi.org/10.1007/978-3-030-37705-2_7

163

164

7 Locally Convex Quasi C*-Algebras and Their Structure

in [0, +∞) coincides with the set of all elements y ∈ A0 , such that y = x ∗ x, for 0 [τ ]+ , is a wedge, but it is not some x ∈ A0 ). Then, as discussed in Sect. 6.2 A necessarily a positive cone; i.e., 0 [τ ]+ ∩ (−A 0 [τ ]+ ) = {0}. A 0 [τ ]+ The set A c is not even, in general, a wedge. But, if A0 is commutative, then of  0 [τ ]+ course, A0 [τ ]+ = A c . • If (A[τ ], A0 ) is a locally convex quasi C*-algebra, in the sense discussed at the beginning of this section, we shall call positive (resp., commutatively positive) an 0 [τ ] and, for element of A, which is positive (resp., commutatively positive) in A simplifying the notations, we shall denote by A+ (resp., A+ ) the corresponding c set. Then, we have τ

0 [τ ]+ = A+ , A+ := A ∩ A 0 τ

denotes, as usual, the closure of A+ where A+ 0 0 in A[τ ]. Further we employ the following two extra conditions (T3 ), (T4 ) for the locally 0 [τ ]+ convex topology τ on A0 and examine the effect on A c . (T3 ) For each λ ∈ , there exists λ ∈ , such that pλ (xy) ≤ x0 pλ (y),

∀ x, y ∈ A0 with xy = yx;

  + (T4 ) The set U (A+ 0 ) := x ∈ A0 : x0 ≤ 1 is τ -closed. Proposition 7.1.2 Let A0 [ · 0 ] be a unital C*-algebra and τ a locally convex 0 [τ ] is a topology on A0 . Suppose that τ fulfils the conditions (T1 )–(T4 ). Then, A locally convex quasi *-algebra over A0 with the properties: −1 0 [τ ]+ 1. for every a ∈ A c , the element e + a is invertible and its inverse (e + a) + belongs to U (A0 ). 0 [τ ]+ 2. For a given a ∈ A c and any ε > 0, let

aε = a(e + εa)−1 .  + Then, {aε }ε>0 is a commuting net in A+ 0 , such that a − aε ∈ A0 [τ ]c and a = τ – lim aε .

ε→0

 + 0 [τ ]+ 3. A c ∩ (−A0 [τ ]c ) = {0}. + +  +  4. If a ∈ A0 [τ ]c and b ∈ A+ 0 , such that b − a ∈ A0 [τ ] , then a ∈ A0 .

7.1 Locally Convex Quasi C*-Algebras

165

Proof + 0 [τ ]+ 1. Let a ∈ A c . Then, there is a net {xα }α∈ in A0 , such that xα xβ = xβ xα , for all α, β ∈  and xα − → a. Using properties of the positive elements of a τ

C ∗ -algebra and the condition (T3 ), we have that for every λ ∈ , there is λ ∈  with



pλ (e + xα )−1 − (e + xβ )−1 = pλ (e + xα )−1 (xα − xβ )(e + xβ )−1 ≤ (e + xα )−1 0 (e + xβ )−1 0 pλ (xα − xβ ) ≤ pλ (xα − xβ ) → 0. So, {(e + xα )−1 }α∈ is a Cauchy net in A0 [τ ] consisting of elements of U (A+ 0 ), which by (T4 ) is τ -closed. Hence, → y ∈ U (A+ (e + xα )−1 − 0 ). τ

(7.1.1)

We shall show that (e + a)−1 exists and equals y. Indeed: Using again condition (T3 ), for each λ ∈ , there is λ ∈  with



pλ e − (e + a)(e + xα )−1 = pλ (xα − a)(e + xα )−1 ≤ (e + xa )−1 0 pλ (xα − a) ≤ pλ (xα − a) → 0. Therefore, → e. (e + a)(e + xα )−1 − τ

(7.1.2)

On the other hand, since xβ y = τ − lim xβ (e + xα )−1 = τ − lim(e + xα )−1 xβ = yxβ , α

α

∀ β ∈ ,

we have ay = ya. Further, we can show that → (e + a)y. (e + a)(e + xα )−1 − τ

(7.1.3)

→ a, for any ε > 0 there exists α0 ∈ , such that for all α ≥ α0 Indeed, since xα − τ

and all λ ∈  one has pλ (xα − a) < ε. Now, by (T3 ) we have that for any α ∈ 

pλ (e + a)(e + xα )−1 − (e + a)y

≤ pλ (e + a)(e + xα )−1 − (e + xα0 )(e + xα )−1

+ pλ (e + xα0 )(e + xα )−1 − (e + xα0 )y + pλ (e + xα0 )y − (e + a)y

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7 Locally Convex Quasi C*-Algebras and Their Structure

≤ pλ (a − xα0 ) + e + xα0 0 pλ (e + xα )−1 − y + pλ (xα0 − a)

< 2ε + e + xα0 0 pλ (e + xα )−1 − y ,

which by (7.1.1) implies that limα pλ (e + a)(e + xα )−1 − (e + a)y = 0. Thus, by (7.1.2) and (7.1.3) we have (e + a)y = e = y(e + a). Hence, (e + a)−1 exists and belongs to U (A+ 0 ) (since y does). 0 [τ ] 2. It is clear from (1) that for every ε > 0 the element (e + εa)−1 exists in A + and belongs to U (A0 ). In particular, applying (T3 ) we get that for each λ ∈ , there is λ ∈  with



−1 pλ e − (e + εa)−1 = εpλ a(e + εa ) ≤ ε(e + εa)−1 0 pλ (a) ≤ εpλ (a), so that τ – lim (e + εa)−1 = e.

(7.1.4)

ε→0

On the other hand, from the very definitions one has



−1 1 aε = a e + εa = (e + εa)−1 a = e − (e + εa)−1 , ε

∀ ε > 0, and



−1 0 [τ ]+ a − aε = a e − (e + εa ) = e − (e + εa)−1 a ∈ A c .

(7.1.5)

Now, by the same way as in (7.1.3), we conclude from (7.1.4) and (7.1.5) that τ – lim aε = a. ε→0

0 [τ ]+  + 3. Let a ∈ A c ∩ (−A0 [τ ]c ). For any ε > 0, we have by (2) that −1 −1 − → a and A+ − → −a. A+ 0 # a(e + εa) 0 # (−a)(e − εa) τ

τ

Thus, if xε := a(e + εa)−1 − (−a)(e − εa)−1 ,

(7.1.6)

we obtain

xε = a (e + εa)−1 + (e − εa)−1 = a(e + εa)−1 (e − εa + e + εa)(e − εa)−1 = 2a(e + εa)−1 (e − εa)−1 , −1 ∈ where by (1) and (2) we conclude that (e − εa)−1 ∈ A+ 0 and a(e + εa) + A+ respectively. Therefore, x ∈ A according to the functional calculus in ε 0 0 commutative C*-algebras. Similarly, we have that

−xε = 2(−a)(e − εa)−1 (e + εa)−1 ∈ A+ 0

7.1 Locally Convex Quasi C*-Algebras

167

since (−a)(e − εa)−1 and (e + εa)−1 belong to A+ 0 . Thus, + xε ∈ A+ 0 ∩ (−A0 ) = {0}

and so (see (7.1.6)) a(e + εa)−1 = −a(e − εa)−1 . Taking τ -limits with ε → 0, we get a = −a, i.e., a = 0. 0 [τ ]+ . 4. By (2) and the assumptions in (4), b − a and a − aε are contained in A + + 0 [τ ] is a wedge, b − aε = (b − a) + (a − aε ) ∈ A 0 [τ ] . Furthermore, Since, A by (T4 ) 0 [τ ]+ ∩ A0 = A+ b − aε ∈ A 0 ,

∀ ε > 0.

Hence, aε 0 ≤ b0 ,

∀ ε > 0,

so that if b = 0, then a = 0 ∈ A+ 0 , since a = τ – lim aε . If b = 0, then ε→0  + + 1 a : ε > 0 ⊂ U (A ) and by (T ), U (A ) is τ -closed; so again we get 4 0 0 b0 ε

that a ∈ A+ 0 .

 

The above lead to the following Definition 7.1.3 Let A0 [ · 0 ] be a unital C*-algebra and τ a locally convex topology on A0 satisfying the conditions (T1 )–(T4 ). If A is a vector subspace 0 [τ ], stable under involution and containing A0 , then we say that the pair of A (A[τ ], A0 ) is a locally convex quasi C*-algebra, or that A[τ ] is a locally convex quasi C*-algebra over A0 . We present now some examples of locally convex quasi C*-algebras. Before we go on we recall the concept of a GB*-algebra. Definition 7.1.4 (Allan) Let A[τ ] be a locally convex *-algebra with unit e and B∗A the collection of all subsets B of A, such that B is absolutely convex, closed   and bounded and moreover e ∈ B, B ⊆ B 2 , and B = B ∗ := a ∗ : a ∈ B . Then, A[τ ] is called a GB*-algebra if the following conditions are fulfilled: 1. The collection B∗A has a greatest member (under the partial ordering of B∗A by inclusion), denoted by B0 ; 2. A[τ ] is symmetric, in the sense that every element of the form e+a ∗ a is invertible for every a ∈ A and its inverse, say b, is a bounded element (i.e., there is λ ∈ C\{0}, such that the set {(λb)n , n ∈ N} is bounded in A[τ ]);

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7 Locally Convex Quasi C*-Algebras and Their Structure

3. A[τ ] is pseudo-complete, in the sense that, for every B ∈ B∗A , the *-subalgebra A[B] of A generated by B is a Banach *-algebra under the norm given by the Minkowski functional of B. All pro C*-algebras (inverse limits of C*-algebras) and all C*-like locally convex *-algebras [63], like for instance, the Arens algebra Lω [0, 1] [41] are GB*-algebras. Example 7.1.5 (GB*-Algebras) Let A[τ ] be a GB*-algebra over B0 . Then, A0 [ · 0 ] ≡ A[B0 ] is a C*-algebra under the C*-norm  · 0 ≡  · B0 given by the Minkowski functional of B0 . Assume that the locally convex topology τ fulfils the condition (T3 ). Then, it is easily checked that (A[τ ], A0 ) is a locally convex quasi C*-algebra over A0 . Example 7.1.6 (Banach Quasi C*-Algebras) Let (A, A0 ) be a normal Banach quasi *-algebra (Definition 4.1.8). In this case, Ab is a Banach *-algebra equipped with the multiplication ∀ a, b ∈ Ab

a • b = La b,

  and the norm ab := max La , R a  , a ∈ Ab (see Corollary 4.1.9). Furthermore, we have (see discussion after (4.1.2)) U (A0 [ · b ])

·

⊂ U (Ab ).

(7.1.7)

·

Indeed, take an arbitrary x ∈ U (A0 [ · b ]) . Then, there is a sequence {xn } in U (A0 [ · b ]), such that lim xn − x = 0. On the other hand, using (7.1.5), we n→∞

have that for each y ∈ A0

xy = lim xn y ≤ lim xn b y ≤ y n

n→∞

and similarly yx ≤ y. Hence, x ∈ U (Ab ). We recall that, if A0 = Ab , then the Banach quasi *-algebra (A[ · ], A0 ) is said to be full. Banach quasi C*-algebras are related to proper CQ*-algebras in the following way: 1. If (A[ · ], A0 ) is a full proper CQ*-algebra, then A[ · ] is a Banach quasi C*-algebra over the C*-algebra A0 [ · b =  · 0 ]. This follows by the very definitions (in this respect, see also Example 7.1.6) and (7.1.4), (7.1.5), (7.1.6). 2. Conversely, suppose that A[ · ] is a Banach quasi C*-algebra over the C*algebra A0 [ · 0 ]. Then, (A[ · ], A0 ) is a proper CQ*-algebra, if and only if, xb = x0 , for all x ∈ A0 .

7.1 Locally Convex Quasi C*-Algebras

169

We consider the following realization of this situation. Let I be a compact interval of R. Then, it is shown that the proper CQ*-algebra (Lp (I ), L∞ (I )) is a Banach quasi C*-algebra over L∞ (I ), but the proper CQ*-algebra (Lp (I ), C(I )) is not a Banach quasi C*-algebra over C(I ). A noncommutative example of a proper CQ∗ -algebra, which is also a Banach quasi C∗ -algebra, is provided by the CQ∗ -algebra (C, (S −1 ) ) introduced in Example 5.3.3. Making use of the weak operator topology of B(H), one can prove that T4 also holds on (S −1 ) . The proof will be given in the next section in a more general context. Then, C is a locally convex quasi C∗ -algebra. Example 7.1.7 In order to describe certain quantum physical models with an infinite number of degrees of freedom, Lassner introduced in [65, 66] some locally convex topologies in a noncommutative C*-algebra that he called physical topologies. In this example we are going to discuss how these topologies can be cast in the framework developed in this chapter. Let A0 [ · 0 ] be a C*-algebra and  = {πα : α ∈ I } a system of *representations of A0 on a dense subspace Dα of a Hilbert space Hα , i.e., each πα is a *-homomorphism of A0 into the O*-algebra L† (Dα ) (see Sect. 7.2). Since A0 [ · 0 ] is a C*-algebra, each πα is a bounded *-representation, i.e., πα (x) ∈ B(Hα ), for every x ∈ A0 . The system  is supposed to be faithful, in the sense that if x ∈ A0 , x = 0, then there exists α ∈ , such that πα (x) = 0. The physical topology τ is the coarsest locally convex topology on A0 , such that every πα ∈  is continuous from A[τ ] into L† (Dα )[τu (L† (Dα ))], where τu (L† (Dα )) is the L† (Dα )-uniform topology of L† (Dα ) (see Sect. 7.2). This topology depends, of course, on the choice of an appropriate system  of *-representations of A0 ; these *-representations are, in general, nothing but the GNS representations constructed starting from a family ωα of states which are relevant (and they are usually called in this way) for the physical model under consideration. Every physical topology satisfies the conditions 0 [τ ] is a T1 , T2 and T4 , but it does not necessarily satisfy T3 . Here we show that A locally convex quasi C*-algebra over A0 for some special choice of the system  of *-representations of A0 . Suppose that Dα = D∞ (Mα ) =



D(Mαn ),

n∈N

where Mα is a selfadjoint unbounded operator. Without loss of generality we may assume that Mα ≥ Iα , with Iα the identity operator in B(Hα ). Let  be a system of *-representations πα of A0 on Dα , such that πα (x)Mα ξ = Mα πα (x)ξ , for every 0 [τ ] is a locally convex quasi C*-algebra x ∈ A0 and for every ξ ∈ Dα . Then, A over A0 . This follows, on the one hand, from the fact that, in this case, the physical topology τ is defined by the family of seminorms   pαf (x) := f (Mα )πα (x), ∀ x ∈ A0 ,

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7 Locally Convex Quasi C*-Algebras and Their Structure

where πα ∈  and f is running in the set F of all positive, bounded and continuous functions on R+ , such that supx∈R+ x k f (x) < ∞, for every k = 0, 1, 2, . . . [66, Lemma 2.8] and, on the other hand, from the inequality     pαf (xy) = f (Mα )πα (x)πα (y) ≤ πα (x)pαf (y), ∀ x, y ∈ A0 .

7.2 Locally Convex Quasi C*-Algebras of Operators Let D be a dense subspace in a Hilbert space H and L† (D, H) the partial *-algebra of operators on D defined in Sect. 2.1.3. Let now M0 [ · 0 ] be a unital C*-algebra over H that leaves D invariant, i.e., M0 D ⊂ D. Then, the restriction M0  D of M0 to D is an O*-algebra on D, therefore an element T of M0 is regarded as an element T  D of M0  D. Moreover, let M0 ⊂ M ⊂ L† (D, H), where M is an O*-vector space on D, that is a *-invariant subspace of L† (D, H). Denote by B(M) the set of all bounded subsets of D[tM ], where tM is the graph topology on M (see discussion after Definition 2.2.5). Further, denote by Bf (D) the set of all finite subsets of D. Then, Bf (D) ⊂ B(M). A subset B of B(M) is called admissible if the following hold: (i) Bf (D) ⊂ B, (ii) ∀ B1 , B2 ∈ B ∃ B3 ∈ B : B1 ∪ B2 ⊂ B3 , (iii) AB ∈ B, ∀ A ∈ M0 and ∀ B ∈ B. It is clear that Bf (D) and B(M) are admissible. Consider now an arbitrary admissible subset B of B(M). Then, for any B ∈ B define the following seminorms on M: pB (T ) := sup |T ξ |η|,

T ∈M

(7.2.8)

ξ,η∈B

pB (T ) := sup T ξ , ξ ∈B

T ∈M

  p†B (T ) := sup T ξ  + T † ξ  , ξ ∈B

(7.2.9) T ∈ M.

(7.2.10)

We call the corresponding locally convex topologies on M defined by the families (7.2.8), (7.2.9) and (7.2.10) of seminorms, B-uniform topology, strongly B-uniform topology, respectively strongly∗ B-uniform topology on M and denote them by tu (B), tu (B), respectively tu∗ (B). In particular, the B(M)-uniform topology, the strongly B(M)-uniform topology, respectively the strongly∗ B(M)-uniform topology will be simply called M-uniform topology, strongly M-uniform topology, respectively strongly∗ M-uniform topology and will be denoted by tu (M), tu (M),

7.2 Locally Convex Quasi C*-Algebras of Operators

171

respectively tu∗ (M). In the book of Schmüdgen [23], these topologies are called bounded topologies and tu (B), tu (B) are denoted by tB , tB , while tu (M), tu (M) are denoted by tD , tD , respectively. The Bf (D)-uniform topology, strongly Bf (D)uniform topology, resp. strongly∗ Bf (D)-uniform topology is called weak topology, strong topology, respectively strong *-topology on M, denoted by tw , ts and ts ∗ , respectively, as we did for the space L† (D, H) in Sect. 2.1.3. All these topologies are related in the following way:

*

*

*

tw & tu (B) & tu (M) (7.2.11)

*

*

*

ts & tu (B) & tu (M) ts ∗ & tu∗ (B) & tu∗ (M).

0 [tu∗ (B)] are locally convex quasi 0 [tu (B)] and M We investigate now whether M C*-algebras over M0 . We must check the properties (T1 )–(T4 ) (stated before and after Definition 7.1.1) for the locally convex topologies tu (B), tu∗ (B) and the operator C*-norm  · 0 on M0 . (T1 ) follows easily for both topologies, since B is admissible and M0 D ⊂ D. For (T2 ) notice that for all T ∈ M0 and B ∈ B we have  

p†B (T ) = sup T ξ  + T † ξ  ≤ 2 sup ξ  T 0 , ξ ∈B

ξ ∈B

so by (7.2.11) we conclude that tu (B) & tu∗ (B) &  · 0 . Concerning t∗u (B), the property (T3 ) follows easily from the very definitions. Now, notice the following: for any T , S ∈ M0 with T S = ST and S ∗ = S, one concludes that  pB (T S) ≤ T 0 sup |S|ξ ξ , ξ ∈B

∀ B ∈ B,

(7.2.12)

where |S| := (S 2 )1/2 . Then, it follows that for any T , S ∈ M0 with T S = ST and S ≥ 0, one has  pB (T S) ≤ T 0 sup Sξ ξ , ξ ∈B

∀ B ∈ B.

We prove (7.2.12). From the polar decomposition of S, there is a unique partial isometry V from H to H, such that S = V |S| = |S|V ,

Ker(V ) = Ker(S) and V S = |S|.

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7 Locally Convex Quasi C*-Algebras and Their Structure

By continuous functional calculus it follows that: T commutes with both |S| and |S|1/2 , but also V |S|1/2 = |S|1/2 V . Thus,     



pB (T S) = sup  T Sξ η  = sup  V |S|T ξ η  ξ,η∈B

ξ,η∈B

   

  = sup  T |S|1/2 ξ |S|1/2 V η  ≤ sup T 0 |S|1/2 ξ |S|1/2 η ξ,η∈B

ξ,η∈B

 2  2 1 T 0 sup |S|1/2 ξ  + |S|1/2 η 2 ξ,η∈B  

≤ T 0 sup |S|ξ ξ , ∀ B ∈ B. ≤

ξ ∈B

But, we can not say whether (T3 ) holds for tu (B). In the case when M0 [ · 0 ] is a von Neumann algebra we have the following: • If M0 is commutative, then (T3 ) holds for the topology tw . • If M is a commutative O*-algebra on D in H, containing M0 , then (T3 ) holds for the topology tu (M). Indeed, suppose that M is commutative with M0 ⊂ M. For each B ∈ B(M) consider the set ) B  := V B : V partial isometry in M0 . Commutativity of M implies that B  ∈ B(M). Moreover, B ⊂ B  . Let now T , S ∈ M0 . Let S = V |S| be the polar decomposition of S. Since M0 is a von Neumann algebra, we have V ∈ M0 , which implies that    



 pB (T S) = sup  T Sξ η  = sup  V T |S|1/2 ξ |S|1/2 η  ξ,η∈B

ξ,η∈B

   ≤ V T 0 sup |S|1/2 ξ  |S|1/2 η ξ,η∈B

 

= T 0 sup |S|ξ ξ ξ ∈B

  = T 0 sup Sξ V ∗ ξ ξ ∈B

   ≤ T 0 sup  Sξ η  = T 0 pB  (S). ξ,η∈B 

Hence, (T3 ) holds for tu (M). (T4 ) This property holds for all topologies in (7.2.11). It suffices to prove (T4 ) for tw the topology tw . So, let T ∈ U (M0 ) be arbitrary. Then, there is a net {Tα } in U (M0 ) with Tα − → T . Notice that the sesquilinear form defined on D × D by tw

 

D × D # (ξ, η) → lim Tα ξ η ∈ C, α

7.2 Locally Convex Quasi C*-Algebras of Operators

173

is bounded. Hence, T can be regarded as a bounded linear operator on H, such that  

  T 0 = 1 and T ξ η = lim Tα ξ η , α

∀ ξ, η ∈ D.

Since D is dense in H, an easy computation shows that    

T x y = lim Tα x y ,



α

∀ x, y ∈ H.

(7.2.13)

This proves that T ∈ M0 ∩ U (B(H)) = U (M0 ), which means that U (M0 ) is tw -closed. A consequence of (7.2.13) is now that U (M+ 0 ) is weakly closed, so that (T4 ) holds for the topology tw on M0 . From (7.2.11), (T4 ) also holds for the topologies tu (B) and tu∗ (B). From the preceding discussion we conclude the following Proposition 7.2.1 Let B be an admissible subset of B(M). Then, both 0 [tu∗ (B)], M0 ) and (M 0 [ts ∗ ], M0 ) are locally convex quasi C*-algebras. If M0 (M is a von Neumann algebra and there is a commutative O*-algebra M on D in H, 0 [tw ], M0 ) and (M 0 [tu (M)], M0 ) are commutative locally containing M0 , then (M convex quasi C*-algebras. Remark 7.2.2 0 [tu (B)] and M 0 [tw ] are locally convex 1. In general, we do not know whether M quasi C*-algebras. 0 [ts ∗ ] over M0 , equals to the completion 2. The locally convex quasi C*-algebra M * [ts ∗ ] of the von Neumann algebra M with respect to the topology ts ∗ , but M 0 0 * [ts ∗ ] is not necessarily a locally convex quasi C*-algebra over M , since in M 0 0 general, M0 D ⊂ D. In the case when M0 D ⊂ D, one has the equality * [ts ∗ ] = M 0 [ts ∗ ], M 0 set-theoretically; but, the corresponding locally convex quasi C*-algebras over M0 do not coincide. In particular, one has that + * 0 [ts ∗ ]+ M c  M0 [ts ∗ ]c .

We present now some properties of the locally convex quasi C*-algebra 0 [ts ∗ ], M0 ). (M 0 [ts ∗ ]+ . Consider the following statements: Proposition 7.2.3 Let A ∈ M 0 [ts ∗ ]+ (i) A ∈ M c ; (ii) (I + A)−1 exists and belongs to U (M+ 0 ); (iii) the closure A of A is a positive selfadjoint operator. Then, one has that (i) ⇒ (ii) ⇒ (iii).

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7 Locally Convex Quasi C*-Algebras and Their Structure

Proof (i) ⇒ (ii) follows from Proposition 7.1.2(1). (ii) ⇒ (iii) Since (I + A)−1 is a bounded selfadjoint operator and (I + A)−1 D ⊂ D, it follows that  

  

(I + A)−1 (I + A∗ )ξ η = I + A∗ )ξ (I + A)−1 η = ξ η, for all ξ ∈ D(A∗ ) and η ∈ D, which implies  

∗  

 A ξ ζ = (I + A∗ )ξ (I + A)−1 (I + A∗ )ζ − ξ ζ  



   = ξ (I + A∗ )ζ − ξ ζ  = ξ A∗ ζ , ∀ ξ, ζ ∈ D(A∗ ). Hence, ξ ∈ D(A) and Aξ = A∗ ξ . It is now easily seen that A is a positive selfadjoint operator.     * [ts ∗ ] and M D ⊂ D. Then, the following Corollary 7.2.4 Suppose that A ∈ M 0

0

statements are equivalent: * [ts ∗ ]+ ; (i) A ∈ M 0 c

(ii) (I + A)−1 ∈ U (M0 )+ ; (iii) A is a positive selfadjoint operator. Proof From Proposition 7.2.3 we have that (i) ⇒ (ii) ⇒ (iii). (iii) ⇒ (i) follows easily by considering the spectral decomposition of A.

 

It is natural now to ask whether there exists an extended C*-algebra (abbreviated to EC*-algebra; see right after for the definition) M on D, such that 0 [ts ∗ ]. M0 ⊂ M ⊂ M   If M is a closed O*-algebra on D in H, let Mb := T ∈ M : T ∈ B(H) be the bounded part of M, where  B(H) is the  C*-algebra of all bounded linear operators on H. Then, when Mb ≡ T : T ∈ Mb is a C*-algebra on H and (I+T ∗ T )−1 ∈ Mb , for each T ∈ M, M is said to be an EC*-algebra on D. For more details on EC*-algebras, see [10, Section 4.3]. The next Proposition 7.2.5 gives a characterization of certain EC*-algebras on D, through the set 0 [ts ∗ ], M0 ). of commutatively positive elements of (M Proposition 7.2.5 Let M be a closed O*-algebra on D, such that M0 ⊂ M ⊂ 0 [ts ∗ ] and Mb = M0 . Then, M is an EC*-algebra on D, if and only if, M+ ⊂ M  M0 [ts ∗ ]+ c . Proof Suppose that M is an EC*-algebra on D and let A ∈ M+ be arbitrary. Then, since Mb = M0 , A is a bounded positive selfdjoint operator with  (I + A)−1 ∈ U (M+ 0 ). But, (M0 [ts ∗ ], M0 ) is a locally convex quasi C*-algebra (Proposition 7.2.1), therefore U (M+ 0 ) is ts ∗ -closed. Note that for each n ∈ N, the elements An := A(I + n1 A)−1 belonging to M+ → A; so 0 , are commuting and An − 0 [ts ∗ ]+ Definition 7.1.1 implies that A ∈ M c .

ts ∗

7.3 Structure of Commutative Locally Convex Quasi C*-Algebras

175

† + 0 [ts ∗ ]+  Conversely, suppose that M+ ⊂ M c . So, A ∈ M implies A A ∈ M0 [ts ∗ ]c , + † −1 therefore (I + A A) ∈ U (M0 ) from Proposition 7.1.2(1). Now, since Mb = M0 we finally get that M is an EC*-algebra on D.  

7.3 Structure of Commutative Locally Convex Quasi C*-Algebras Throughout this section (A[τ ], A0 ) is a commutative locally convex quasi C*algebra, where the C*-algebra A0 is supposed to have a unit element. If the multiplication of A0 with respect to the topology τ is jointly continuous, then A[τ ] is a commutative GB*-algebra [59, Theorem 2.1], and so A[τ ] is isomorphic to a *algebra of C∗ -valued continuous functions on a compact space, which take the value ∞ on at most a nowhere dense subset [33, Theorem 3.9]; C∗ denotes the extended complex plane in its usual topology as the one-point compactification of C. The purpose of this section is to consider a generalization of the above result in the case when the multiplication of A0 [τ ] is not jointly continuous. As a ∗ a is not necessarily defined for a ∈ A[τ ], it is impossible to extend any nonzero multiplicative linear functional ϕ on A0 to A[τ ], like in the case of [32, Proposition 6.8]. Here we show that ϕ is extendable to a C∗ -valued partial multiplicative linear functional ϕ  on τ (see beginning of Sect. 6.2 and discussion before Proposition 7.1.2) and A+ = A + 0 that A+ is isomorphic to a wedge of C∗ -valued positive functions on a compact space, which take the value ∞ on at most a nowhere dense subset. This result will be applied in Sect. 7.4 for studying a functional calculus for positive elements. Consider now the set   M(A0 , A+ ) := ax + y : a ∈ A+ , x, y ∈ A0 and denote by M(A0 ) the Gelfand space of A0 (see also beginning of Sect. A.6.1), i.e., the set of all nonzero multiplicative linear functionals on A0 , endowed with the weak∗ -topology σ (M(A0 ), A0 ). Now, let a ∈ A+ and x, y ∈ A0 . Suppose x is hermitian (i.e., x ∗ = x). Then, by continuous functional calculus, x is uniquely decomposed in the following way: x = x+ − x− , x+ , x− ∈ A+ 0 , x+ x− = 0 |x| ≡ (x ∗ x)1/2 = x+ + x− ∈ A+ 0 . Hence, a|x|, ax+ , ax− ∈ A+ , and by (1) and (2) of Proposition 7.1.2, (e + a|x|)−1 , a|x|(e + a|x|)−1 ∈ A+ 0 . Furthermore, since a|x|(e + a|x|)−1 − ax+ (e + a|x|)−1 = ax− (e + a|x|)−1 ∈ A+ ,

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7 Locally Convex Quasi C*-Algebras and Their Structure

Proposition 7.1.2(4) implies that ax+ (e + a|x|)−1 ∈ A+ 0 . Similarly, ax− (e + a|x|)−1 ∈ A+ . Hence, we have that the element 0 (ax + y)(e + a|x|)−1 = ax+ (e + a|x|)−1 − ax− (e + a|x|)−1 + y(e + a|x|)−1 belongs to A0 . Since a general element x of A0 is a linear combination of two hermitian elements of A0 , we finally obtain that (ax + y)(e + a|x|)−1 ∈ A0 ,

∀ a ∈ A+ and x, y ∈ A0 .

Indeed, let x be arbitrary in A0 . Then, x = x1 + ix2 , with x1 and x2 hermitian. An easy computation shows that |x| ≤ |x1 | + |x2 |, |xj | ≤ |x| and (e + a|xj |)(e + a|x|)−1 ≤ e, j = 1, 2. The latter together with Proposition 7.1.2(4) gives (e + a|xj |)(e + a|x|)−1 ∈ A+ 0 ; moreover, from the above (axj + y)(e + a|xj |)−1 ∈ A0 . Thus, for j = 1, 2, we obtain



(axj + y)(e + a|x|)−1 = (axj + y)(e + a|xj |)−1 (e + a|xj |)(e + a|x|)−1 ∈ A0 , which implies (ax + y)(e + a|x|)−1 = (ax1 + y)(e + a|x|)−1 + iax2 (e + a|x|)−1 ∈ A0 .

Hence, the elements ϕ((e + a|x|)−1 ) and ϕ (ax + y)(e + a|x|)−1 are complex numbers, for each ϕ ∈ M(A0 ), so that we can consider the correspondence ϕ  : M(A0 , A+ ) −→ C∗ ≡ C ∪ {∞}, with ⎧

−1 ⎪ ⎨ ϕ (ax+y)(e+a|x|)

, ϕ (e+a|x|)−1 ax + y → ϕ  (ax + y) = ⎪ ⎩∞,



if ϕ (e + a|x|)−1 = 0

if ϕ (e + a|x|)−1 = 0.

Then, we have Lemma 7.3.1 The following statements hold: 1. For every ϕ ∈ M(A0 ) the correspondence ϕ  , given above, is well-defined. 2. Let a ∈ A+ and x ∈ A0 . Then, (e + a)−1 exists in A0 (from Proposition 7.1.2(1)) and we have:



(i) ϕ (e + a|x|)−1 = 0 implies ϕ (e + a)−1 = 0, ϕ ∈ M(A0 );



(ii) ϕ (e + a)−1 = 0 and ϕ(x) = 0 imply ϕ (e + a|x|)−1 = 0, ϕ ∈ M(A0 ).

7.3 Structure of Commutative Locally Convex Quasi C*-Algebras

177

Proof 1. Let a, b ∈ A+ and x, y, z, w ∈ A0 , such that ax + y = bz + w. Then, for every ϕ ∈ M(A0 ) one has that



ϕ (e + a|x|)−1 = 0 ⇔ ϕ (e + b|z|)−1 = 0. (7.3.14) Indeed, we first show (7.3.14) in case x and z are hermitian. Since ax + y = bz + w, we have (e + a|x|) − 2ax− + y = (e + b|z|) − 2bz− + w. We multiply the last equality by (e + a|x|)−1 (e + b|z|)−1 and obtain (e + b|z|)−1 − 2ax− (e + a|x|)−1 (e + b|z|)−1 + y(e + a|x|)−1 (e + b|z|)−1 = (e + a|x|)−1 − 2bz− (e + b|z|)−1 (e + a|x|)−1 + w(e + a|x|)−1 (e + b|z|)−1 . This implies that for every ϕ ∈ M(A0 )



ϕ (e + a|x|)−1 = 0 ⇔ ϕ (e + b|z|)−1 = 0.

(7.3.15)

We next prove (7.3.14) in the case when x and z are arbitrary elements of A0 . Then, the elements x, y, z and w are decomposed into x = x1 + ix2 ,

y = y1 + iy2 ,

z = z1 + iz2 ,

w = w1 + iw2 ,

where xj , yj , zj , wj (j = 1, 2) are hermitian elements in A0 that satisfy the equations: ax1 + y1 = bz1 + w1 ,

ax2 + y2 = bz2 + w2 .

We show now that



ϕ (e + a|x|)−1 = 0 ⇔ either ϕ (e + a|x1 |)−1 = 0

or ϕ (e + a|x2 |)−1 = 0.

(7.3.16)

(7.3.17)



Suppose that ϕ (e + a|x1 |)−1 = 0 and ϕ (e + a|x2 |)−1 = 0. Then, inserting

the quantity e + a(|x1 | + |x2 |))−1 (e + a|x1 | + a|x2 |) in front of the second term of the left hand-side equality in (7.3.18) and doing calculations by repeating this pattern, you obtain the right hand-side of the equality that follows:

−1 e + a(|x1 | + |x2 |) − (e + a|x1 |)−1 (e + a|x2 |)−1





−1 a|x1 |(e + a|x1 |)−1 a|x2 |(e + a|x2 |)−1 ∈ A+ = e + a(|x1 | + |x2 |) 0 , (7.3.18)

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7 Locally Convex Quasi C*-Algebras and Their Structure

whence



ϕ (e + a(|x1 | + |x2 |))−1 ≥ ϕ (e + a|x1 |)−1 (e + a|x2 |)−1 > 0. Furthermore, since |x| ≤ |x1 | + |x2 |, we have



0 < ϕ (e + a(|x1 | + |x2 |))−1 ≤ ϕ (e + a|x|)−1 .



Therefore, ϕ (e + a|x|)−1 = 0. Conversely, suppose that ϕ (e + a|x1 |)−1 = 0

or ϕ (e + a|x2 |)−1 = 0. Then, since (e + a|xj |)−1 ≥ (e + a|x|)−1 , j = 1, 2,

we have that ϕ (e + a|x|)−1 = 0. Now from (7.3.15), (7.3.16) and (7.3.17) we get





ϕ (e + a|x|)−1 = 0 ⇔ ϕ (e + a|x1 |)−1 = 0 or ϕ (e + a|x2 |)−1 = 0



⇔ ϕ (e + b|z1 |)−1 = 0 or ϕ (e + b|z2 |)−1 = 0

⇔ ϕ (e + b|z|)−1 = 0. Thus, (7.3.14) has been shown. Furthermore, by assumption ax + y = bz + w, consequently ϕ  (ax + y) = ∞ ⇔ ϕ  (bz + w) = ∞. On the other hand, from (7.3.14) it follows that ϕ  (ax + y) < ∞ ⇔ ϕ  (bz + w) < ∞. In this case,

ϕ (ax + y)(e + a|x|)−1 (e + b|z|)−1



= ϕ  (bz + w) ϕ (ax + y) = ϕ (e + a|x|)−1 ϕ (e + b|z|)−1 

and this completes the proof of (1). 2. (i) Suppose ϕ (e + a|x|)−1 = 0, ϕ ∈ M(A0 ), Then, (e + a)−1 = (e + a|x|)−1 (e + a|x|)(e + a)−1

= (e + a|x|)−1 (e + a)−1 + a|x|(e + a)−1

= (e + a|x|)−1 (e + a)−1 + |x| − |x|(e + a)−1

= (e + a|x|)−1 (e − |x|)(e + a)−1 + |x| , where (e−|x|)(e+a)−1 +|x| ∈ A0 . So applying ϕ we have ϕ((e+a)−1 ) = 0.

7.3 Structure of Commutative Locally Convex Quasi C*-Algebras

179

(ii) Suppose that ϕ (e + a)−1 = 0 and ϕ(x) = 0, ϕ ∈ M(A0 ). Then, we apply ϕ to the final result of the preceding calculation in (i) and we take

ϕ (e + a|x|)−1 ϕ(|x|) = 0.

Since ϕ(x) = 0, if and only if, ϕ(|x|) = 0, clearly we have ϕ (e + a|x|)−1   = 0. Proposition 7.3.2 For ϕ ∈ M(A0 ), the well defined map ϕ  has the following properties: (1) ϕ  ⊃ ϕ (i.e., ϕ  is an extension of ϕ); (2) ϕ  (ax + y) = ϕ  (a)ϕ(x) + ϕ(y) and ϕ  (ax) = ϕ  (a)ϕ(x), whenever a ∈ A+ and x, y ∈ A0 , such that ϕ  (a)ϕ(x) = ∞ · 0; (3) ϕ  (a + b) = ϕ  (a) + ϕ  (b), for all a, b ∈ A+ ; (4) ϕ  (λa) = λϕ  (a), for all λ ∈ C and a ∈ A+ , where 0 · ∞ = 0. Proof 1. is trivial. 2. Suppose that ϕ  (a)ϕ(x) = ∞ · 0, ϕ ∈ M(A0 ). Then, from the definition of ϕ  and Lemma 7.3.1(2), we have the following implications (considering separately the cases where ϕ  (a) is infinite or not): •

•

ϕ  (ax + y) = ∞ ⇔ ,

ϕ (e + a|x|)−1 = 0

ϕ  (a) = ∞ ,

ϕ (e + a)−1 = 0 ⇓ ϕ  (a)ϕ(x) + ϕ(y) = ∞.

ϕ  (ax + y) < ∞ ⇔ ϕ  (a) < ∞ , ,



−1 = 0 ϕ (e + a)−1 = 0. ϕ (e + a|x|)

So, in this case we also obtain

ϕ ax(e + a|x|)−1 

ϕ (ax + y) = + ϕ(y) ϕ (e + a|x|)−1

ϕ a(e + a)−1 ϕ(x)

+ ϕ(y) = ϕ  (a)ϕ(x) + ϕ(y), = ϕ (e + a)−1 and this completes the proof of (2). (3) Observe that for any a, b ∈ A+ , one has (e + a)−1 (e + b)−1 = (e + a + b)−1 (e + a)−1 (e + b)−1

+ a(e + a)−1 (e + b)−1 + (e + a)−1 b(e + b)−1 ,

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7 Locally Convex Quasi C*-Algebras and Their Structure

where (e + a)−1 (e + b)−1 + a(e + a)−1 (e + b)−1 + (e + a)−1 b(e + b)−1 ∈ A0 (see Proposition 7.1.2). Thus, applying any ϕ ∈ M(A0 ) to the last equality we conclude that



ϕ (e + a + b)−1 = 0 implies either ϕ (e + a)−1 = 0 (7.3.19) or ϕ((e + b)−1 ) = 0. Conversely, observe that (e + a)−1 = (e + a + b)−1 + b(e + a + b)−1 (e + a)−1 , where b(e + a + b)−1 ∈ A+ 0 by Proposition 7.1.2(4), since (a + b)(e + a + b)−1 − a(e + a + b)−1 = b(e + a + b)−1 ∈ A+ with (a + b)(e + a + b)−1 ∈ A+ 0 . So, taking also into account an analogous equality for (e + b)−1 , as well as (7.3.19) we have that

ϕ (e + a + b)−1 = 0 ⇔





either ϕ (e + a)−1 = 0 or ϕ (e + b)−1 = 0,

for all ϕ ∈ M(A0 ). Using now the preceding equivalence, clearly we conclude that: • ϕ  (a + b) = ∞ ⇔

either ϕ  (a) = ∞ or ϕ  (b) = ∞; thus,

ϕ  (a + b) = ϕ  (a) + ϕ  (b) = ∞; or • ϕ  (a + b) < ∞ ⇔ ϕ  (a) < ∞ and ϕ  (b) < ∞. In this case, ϕ  (a + b)

ϕ a(e + a)−1 (e + b)−1 (e + a + b)−1 + b(e + a)−1 (e + b)−1 (e + a + b)−1





= ϕ (e + a)−1 ϕ (e + b)−1 ϕ (e + a + b)−1

ϕ b(e + b)−1 ϕ a(e + a)−1

+

= ϕ (e + a)−1 ϕ (e + b)−1 = ϕ  (a) + ϕ  (b).

(4) It follows from (2) by replacing x with λe, λ ∈ C, and y with 0.

 

7.3 Structure of Commutative Locally Convex Quasi C*-Algebras

181

Remark 7.3.3 In order to have all the values of ϕ  fully determined, we need to define the following: • ϕ  (a)ϕ(x), ϕ  (ax) + ϕ  (bx) and ϕ  (a)ϕ(x1 ) + ϕ  (a)ϕ(x2 ), for any a, b ∈ A+ and x1 , x2 ∈ A0 . From Proposition 7.3.2 we conclude that: +  (i) ϕ  (a)ϕ(x) = ϕ  (ax), for any a ∈ A and x ∈ A0 with ϕ (a)ϕ(x) = ∞ · 0. (ii) ϕ  (ax) + ϕ  (bx) = ϕ  (a + b)x , for any a, b ∈ A+ and x ∈ A0 with either ϕ  (a)ϕ(x) = ∞ · 0 or ϕ  (b)ϕ(x) = ∞ · 0.  + (iii) ϕ  (a)ϕ(x 1 + x 2 ) = ϕ a(x1 + x2 ) , for any a ∈ A and x1 , x2 ∈ A0 with ϕ  a(x1 + x2 ) = ∞ · 0.

Furthermore, the definition of ϕ  and Proposition 7.3.2 imply that: 1. When ϕ  (a) = ∞ and ϕ(x) = 0, the value ϕ  (ax) of ϕ  depends upon a and x. For instance, • x = 0 ⇒ ϕ  (ax) = ϕ  (0) = ϕ(0)

= 0;



• x = (e + a)−1 ⇒ ϕ  a(e + a)−1 = ϕ a(e + a)−1 = ϕ e − (e + a)−1 = 1. 2. For a, b ∈ A+ and x ∈ A0 , such that either ϕ  (a)ϕ(x) = ∞ · 0 or ϕ  (b)ϕ(x) = ∞ · 0, the value ϕ  ((a + b)x) clearly depends upon a, b and x. 3. For a ∈ A+ and x1 , x2 ∈ A0 , such that either ϕ  (a)ϕ(x 1 ) = ∞ · 0 or

ϕ  (a)ϕ(x2 ) = ∞ · 0, then again the value ϕ  a(x1 + x2 ) depends upon a, x1 and x2 . Conclusion We define the requested values of ϕ  by (i), (ii) and (iii), for any a, b ∈ A+ and x1 , x2 ∈ A0 . Remark 7.3.4 We do not know whether ϕ  is defined or not on the linear span of M(A0 , A+ ). Now, for any a ∈ A+ and x, y ∈ A0 , we fix the notation:  ax + y(ϕ) ≡ ϕ  (ax + y),

ϕ ∈ M(A0 ).

Then, we have the following  Proposition 7.3.5 With a, x, y, ϕ as before, ax + y is a C∗ -valued continuous function on the compact Hausdorff space M(A0 ), which takes the value ∞ on at most a nowhere dense subset of M(A0 ). Proof We shall show that the set    Nax+y + y(ϕ) = ∞ ,  ≡ ϕ ∈ M(A0 ) : ax is a nowhere dense closed subset of M(A0 ). Notice that

  −1 =0 , Nax+y  = ϕ ∈ M(A0 ) : ϕ (e + a|x)

(7.3.20)

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7 Locally Convex Quasi C*-Algebras and Their Structure

from which it follows that Nax+y  is closed. Now, suppose that ∃ U non-empty open subset of M(A0 ) with U ⊂ Nax+y . From the commutative Gelfand–Naimark theorem, A0 = C(M(A0 )), with respect to an isometric *-isomorphism. Thus, using Urysohn’s lemma for M(A0 ) we get that b(ϕ) = ϕ(b) = 0, ∃ b ∈ A0 : b0 = 1 and 

∀ ϕ ∈ U .

But this together with (7.3.20) and the fact that U ⊂ Nax+y  , implies

ϕ b(e + a|x|)−1 = 0,

∀ ϕ ∈ M(A0 ).

The afore-mentioned identification A0 = C(M(A0 )) gives now b(e + a|x|)−1 = 0, which clearly yields b = 0, a contradiction to b0 = 1. Hence, Nax+y  is a nowhere dense closed subset of M(A0 ).  Next we show that ax + y is continuous on M(A0 ). Put z ≡ (e + a|x|)−1 and w ≡ ax(e + a|x|)−1 . Take an arbitrary ϕ0 ∈ M(A0 ) and consider the cases:  • ax + y(ϕ0 ) = ∞, i.e.,  z(ϕ0 ) = 0. From the continuity of  z there is a neighborhood Uϕ0 of ϕ0 with  z(ϕ) = 0, for all ϕ ∈ Uϕ0 . Thus, we get  ax + y(ϕ) =

w (ϕ) + y (ϕ),  z(ϕ)

∀ ϕ ∈ U ϕ0 ,

where all functions w , z,  y are continuous at ϕ0 , so that the same is true for  ax + y.  • ax + y(ϕ0 ) = ∞, i.e.,  z(ϕ0 ) = 0. Take an arbitrary net {ϕα } in M(A0 ), such that ϕα → ϕ0 , with respect to the weak*topology σ (M(A0 ), A0 ). Then, z(ϕ0 ) = 0,  z(ϕα ) →  where  z(ϕα ) = 0, since Nax+y  is a nowhere dense subset of M(A0 ). Since |" ax(ϕα )| =



1/2 ϕα (ax(e + a|x|)−1 )(ax(e + a|x|)−1 )

ϕα (e + a|x|)−1

7.3 Structure of Commutative Locally Convex Quasi C*-Algebras



1/2 ϕα (a(e + a|x|)−1 )(x ∗ xa(e + a|x|)−1 )

ϕα (e + a|x|)−1

1/2 ϕα (a|x|(e + a|x|)−1 )2

ϕα (e + a|x|)−1

ϕα a|x|(e + a|x|)−1

ϕα (e + a|x|)−1

ϕα e − (e + a|x|)−1

ϕα (e + a|x|)−1

= = = = =

183

1 − 1,  z(ϕα )

x(ϕα ) = ∞, which implies it follows that lim a" α

  + y(ϕα ) = ∞ = ax + y(ϕ0 ). lim ax α

This completes the proof.

 

All the above lead to the following Definition 7.3.6 Let W be a completely regular topological space and F(W )+ the set of all C∗ -valued positive continuous functions on W , which take the value ∞ on at most a nowhere dense subset W0 of W . Then, F(W )+ is said to be a wedge on W , if for any f, g ∈ F(W )+ and λ ≥ 0, the functions f + g and λf defined pointwise on W0 , on which f and g are both finite, are extendible to C∗ -valued positive continuous functions on W that also belong to F(W )+ . We keep the same symbols f + g and λf for the respective extensions. Consider now the set   F(W ) ≡ f g0 + h0 : f ∈ F(W )+ , g0 , h0 ∈ C(W ) , where C(W ) is the *-algebra of all continuous C-valued functions on W . Then, the set F(W ) fulfils the following conditions: • (f1 + f2 )g0 = f1 g0 + f2 g0 , • (λf )g0 = λ(fg0 ), • f (g0 + h0 ) = f g0 + f h0 , for all f, f1 , f2 ∈ F(W )+ , g0 , h0 ∈ C(W ) and λ ≥ 0. Definition 7.3.7 We call F(W ) the set of C∗ -valued positive continuous functions on W generated by the wedge F(W )+ and the *-algebra C(W ).

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7 Locally Convex Quasi C*-Algebras and Their Structure

In this regard (see also Remark 7.3.3), we have the following   Theorem 7.3.8 Let F(M(A0 ))+ ≡  a : a ∈ A+ . Then, 1. F(M(A0 ))+ is a wedge on M(A0 ).  2. The map : M(A0 , A+ ) → F(M(A0 )) : ax + y → ax + y, is a bijection satisfying the properties: (i) (A+ ) = F(M(A0 ))+ , with (a + b) = (a) + (b) and (λa) = λ (a), for all a, b ∈ A+ and λ ≥ 0. (ii) (A0 ) = C(M(A0 )), being an isometric *-isomorphism from A0 onto C(M(A0 )).

(iii) (ax) a ∈ A+ and x ∈ A0 . (a + b)x = (a) +

= (a) (x), for all +

(b) (x), for all a, b ∈ A and x ∈ A0 .

(λax) = λ (a) (x), for

all a ∈ A+ , x ∈ A0 and λ ≥ 0. a(x1 + x2 ) = (a) (x1 ) + (x2 ) , for all a ∈ A+ and x1 , x2 ∈ A0 . Proof The statements (1), (2)(i) and (2)(ii) follow from Propositions 7.3.2 and 7.3.5. We show the statement (2)(iii). Let a ∈ A+ and x ∈ A0 . From Proposition 7.3.5,  a and a" x are C∗ -valued continuous functions on M(A0 ) that take the value ∞ on at most a nowhere dense subset of M(A0 ). Hence, the set   K ≡ ϕ ∈ M(A0 ) :  a (ϕ) < ∞ and a" x(ϕ) < ∞ is dense in M(A0 ) and a" x(ϕ) =  a (ϕ) x (ϕ), ∀ ϕ ∈ K, therefore by the continuity of  a and a" x we conclude that a" x = a x , from which it follows that (ax) = (a) (x). The rest of the properties in (2)(iii) are similarly proved.  

7.4 Functional Calculus for Positive Elements Throughout this section (A[τ ], A0 ) is a commutative locally convex quasi C*algebra with unit e ∈ A0 . Here we shall consider a functional calculus for the positive elements of A[τ ], resulting, for instance, to consideration of the nth-root of an element a ∈ A+ (see Corollary 7.4.7). For this purpose, we first need to extend the multiplication of A[τ ]. Definition 7.4.1 Let a, b ∈ A+ ; a is called left-multiplier of b, and we write a ∈ L(b), if there exist nets {xα }, {yβ } in A+ → a, yβ − → b and xα yβ − →c 0 , such that xα − τ τ τ (in the sense that the double indexed net {xα yβ } converges to c). The product of a, b denoted by ab is given as follows ab := c = τ − lim xα yβ . α,β

7.4 Functional Calculus for Positive Elements

185

Lemma 7.4.2 The product ab is well-defined, in the sense that it is independent of the selection of the nets {xα }, {yβ }. Proof Let {xα }, {yβ } be two nets in A+ 0 , such that → a, yβ − → b and xα yβ − → c. xα − τ

τ

τ

Then, (see also Proposition 7.1.2) (e + xα )−1 xα yβ (e + yβ )−1 (e + c)−1 − (e + a)−1 c(e + c)−1 (e + b)−1

= (e + xα )−1 xα yβ (e + yβ )−1 (e + c)−1 − (e + xα )−1 c(e + c)−1 (e + yβ )−1

+ (e + xα )−1 c(e + c)−1 (e + yβ )−1 − (e + a)−1 c(e + c)−1 (e + yβ )−1

+ (e + a)−1 c(e + c)−1 (e + yβ )−1 − (e + a)−1 c(e + c)−1 (e + b)−1 . → a, so taking As we have seen in the proof of Proposition 7.1.2, (1) (e + xα )−1 − τ τ -limits in the preceding equality, we conclude that → (e + a)−1 c(e + c)−1 (e + b)−1 . (e + xα )−1 xα yβ (e + yβ )−1 (e + c)−1 − τ

On the other hand,



(e + xα )−1 xα yβ (e + yβ )−1 (e + c)−1 − (e + a)−1 a b(e + b)−1 (e + c)−1

= (e + xα )−1 xα − (e + a)−1 a yβ (e + yβ )−1 (e + c)−1

+ (e + a)−1 a yβ (e + yβ )−1 − b(e + b)−1 (e + c)−1 , from which, as before, we take that

→ (e + a)−1 a b(e + b)−1 (e + c)−1 . (e + xα )−1 xα yβ (e + yβ )−1 (e + c)−1 − τ

Hence, we finally obtain



(e + a)−1 c(e + b)−1 = (e + a)−1 a b(e + b)−1 . Suppose now that two other nets {xλ }, {yμ } exist in A+ 0 , such that → a, yμ − → b and xλ yμ − → c . xλ − τ

τ

τ

Working exactly as before we are led to the equality



(e + a)−1 c (e + b)−1 = (e + a)−1 a b(e + b)−1 ,

(7.4.21)

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7 Locally Convex Quasi C*-Algebras and Their Structure

which together with (7.4.21) gives (e + a)−1 c(e + b)−1 = (e + a)−1 c (e + b)−1 ⇔ c = c .

 

We may now set the following Definition 7.4.3 Let a, b ∈ A+ with a ∈ L(b) and x, y ∈ A0 . The product of the elements ax, by is defined as follows: (ax)(by) := (ab)xy. Further, we consider the spectrum of an element a ∈ A+ . Definition 7.4.4 Let a ∈ A+ . The spectrum of a denoted by σA0 (a), is that subset of C∗ , defined in the following way: • Let λ ∈ C. Then, λ ∈ σA0 (a) ⇔ λe − a has no inverse in A0 ; • ∞ ∈ σA0 (a) ⇔ a ∈ A0 . Lemma 7.4.5 Let a ∈ A+ . Then,     a (ϕ) : ϕ ∈ M(A0 ) ⊂ R+ ∪ ∞ . σA0 (a) =  In particular, σA0 (a) is a locally compact subset of C∗ . Proof Let λ ∈ C. Then, (see also Theorem 7.3.8) λ ∈ σA0 (a) ⇔ (λe − a)−1 ∈ A0 ⇔ λ =  a (ϕ), ∀ ϕ ∈ M(A0 ). Let now λ = ∞. Then,

a ∈ C M(A0 ) ⇔ ∃ ϕ0 ∈ M(A0 ) :  a (ϕ0 ) = ∞. λ ∈ σA0 (a) ⇔ a ∈ A0 ⇔   

If a ∈ denote by Cb σA0 (a) , the C*-algebra of all bounded continuous

functions on σA0 (a). For n ∈ N and f ∈ C σA0 (a) , define the function The rest is clear. A+ ,

gn (λ) :=

f (λ) , (1 + λ)n

λ ∈ σA0 (a).

(7.4.22)

In this regard, set



Cn σA0 (a) := {f ∈ C σA0 (a) ∩ R : gn ∈ Cb σA0 (a) }. Then,

Cb σA0 (a) ⊂ C1 σA0 (a) ⊂ C2 σA0 (a) ⊂ · · · .

(7.4.23)

7.4 Functional Calculus for Positive Elements

187

Now, the promised functional calculus for positive elements in A[τ ] is given by the following Theorem 7.4.6 Let a ∈ A+ . Suppose that the element a n is well-defined for some n

, n ∈ N. Then, there is a unique *-isomorphism f → f (a) from Ck σA0 (a) into k=1

A[τ ], in such a way that: (i) If u0 ∈

n , k=1



Ck σA0 (a) , with u0 (λ) = 1, for each λ ∈ σA0 (a), then u0 (a) =

e ∈ A0 → A[τ ]. n

, Ck σA0 (a) , with u1 (λ) = λ, for each λ ∈ σA0 (a), then u1 (a) = (ii) If u1 ∈ k=1

a ∈ A[τ ]. n

, Ck σA0 (a) and ϕ ∈ M(A0 ); (iii) f(a)(ϕ) = f ( a (ϕ)), for any f ∈ k=1

(iv) (f1 + f2 )(a) = f1 (a) + f2 (a), for any f1 , f2 ∈ n ,

n , k=1



Ck σA0 (a) , (λf )(a) =



Ck σA0 (a) and λ ∈ C, (f1 f2 )(a) = f1 (a)f2 (a), for

λf (a), for any f ∈ k=1

any fj ∈ Ckj σA0 (a) , j = 1, 2, with k1 + k2 ≤ n.

(v) Restricted to Cb σA0 (a) the map f → f (a) is an isometric *-isomorphism of the C*-algebra Cb σA0 (a) onto the closed *-subalgebra of the C*-algebra A0 generated by e and (e + a)−1 .

Ck σA0 (a) . Then, f ∈ Ck σA0 (a) , for some k with 1 ≤ k ≤ k=1

f (λ) n, and gk ∈ Cb σA0 (a) with gk (λ) := (1+λ) k , λ ∈ σA0 (a). From Lemma 7.4.5

a ∈ C M(A0 ) , therefore (Gelfand–Naimark theorem) there is a we have that gk ◦  unique element gk (a) ∈ A0 , such that Proof Let f ∈

n ,



g a (ϕ) , k (a)(ϕ) = gk 

∀ ϕ ∈ M(A0 ).

(7.4.24)

Now let f (a) := gk (a)(e + a)k ∈ A[τ ].

(7.4.25)

We shall show that f (a) does not depend on k, 1 ≤ k ≤ n. Indeed, let f ∈ Cj (σA0 (a)) with: • j ≤ k; then, for each λ ∈ σA0 (a), gk (λ) =

1 f (λ) f (λ) 1 = = gj (λ) . (1 + λ)k (1 + λ)j (1 + λ)k−j (1 + λ)k−j

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7 Locally Convex Quasi C*-Algebras and Their Structure

Hence, gk (a) = gj (a)(e + a)−(k−j ) ∈ A0 and gk (a)(e + a)k = gj (a)(e + a)j ;

(7.4.26)

• j > k; in this case too, one takes (7.4.26) in a similar way. So, the element f (a) ∈ A[τ ] is well-defined by (7.4.25). Now, it is easily seen that the map f → f (a) from

n )



Ck σA0 (a) into A[τ ]

k=1

is a *-isomorphism with the properties (i), (ii), (iii).

(iv) Consider the functions f1 ∈ Ck1 σA0 (a) , f2 ∈ Ck2 σA0 (a) with k1 + k 2 ≤ n. Then, (see (7.4.23) and discussion before (7.4.24)) gki ∈ Cb σA0 (a) with gki (a) unique in A0 , i = 1, 2. Define

the function f (λ) := f1 (λ)f2 (λ), λ ∈ σA0 (a). Then, f ∈ Ck1 +k2 σA0 (a) and gk1 +k2 (λ) =

f (λ) = gk1 (λ)gk2 (λ), (1 + λ)k1 +k2

λ ∈ σA0 (a),



that is gk1 +k2 ∈ Cb σA0 (a) . Hence, gk1 +k2 (a) = gk1 (a)gk2 (a) ∈ A0 . Moreover (see also Definition 7.4.3 and (7.4.25)) (f1 f2 )(a) = f (a) = gk1 +k2 (a)(e + a)k1 +k2



= gk1 (a)(e + a)k1 gk2 (a)(e + a)k2 = f1 (a)f2 (a). The first two equalities in (iv) are similarly shown. (v) Arguing as in (7.4.24) and taking into account Lemma 7.4.5, we easily reach   at the conclusion. Corollary 7.4.7 Let a ∈ A+ and n ∈ N. Then, there is unique b ∈ A+ , such that 1 a = bn . The element b is called nth-root of a and is denoted by a n . If, in particular, 1 n = 2, the element a 2 is called square-root of a. 1

1

1− n n Proof Consider the functions , λ ≥ 0, which

f1 (λ) := λ and f2 (λ) := λ clearly belong to C1 σA0 (a) . Then, (see (7.4.22), (7.4.23)) g1 , g2 ∈ Cb σA0 (a) with g1 (λ) = f1 (λ)(1 + λ)−1 , g2 (λ) = f2 (λ)(1 + λ)−1 , λ ≥ 0. Theorem 7.4.6 gives that the elements f1 (a), f2 (a) are uniquely defined in A[τ ] with

f1 (a) = g1 (a)(e + a),

f2 (a) = g2 (a)(e + a),

7.5 Structure of Noncommutative Locally Convex Quasi C*-Algebras

189

where gi (a) ∈ A+ 0 , i = 1, 2 (see, e.g., (7.4.24)). Moreover (see also Proposition 7.1.2, (1) and (2)), for each ε > 0 ε→0

−1 A+ −−→ f1 (a), resp. 0 # g1 (a)(e + a)(e + εa) τ

ε→0

−1 −−→ f2 (a). A+ 0 # g2 (a)(e + a)(e + εa) τ

On the other hand, since (f1 f2 )(λ) = λ, from Theorem 7.4.6(ii) we have that (f1 f2 )(a) = a, therefore (see also Proposition 7.1.2(2))



ε→0 g1 (a)(e + a)(e + εa)−1 g2 (a)(e + a)(e + εa)−1 = a(e + εa)−1 −−→ a. τ

So, from Definition 7.4.1, we conclude that

f1 (a) ∈ L f2 (a) and a = f1 (a)f2 (a). Now, since f2 (a) ∈ A+ , we repeat the previous procedure with f2 (a) in the place of a, so that continuing in this way we finally obtain a = f1 (a)f1 (a) · · · f1 (a)

(n-times).

The proof is completed by taking b = f1 (a).

 

7.5 Structure of Noncommutative Locally Convex Quasi C*-Algebras In this section we consider a noncommutative locally convex quasi C*-algebra (A[τ ], A0 ), with unit e ∈ A0 and we investigate the following: (a) Conditions under which such an algebra is continuously embedded in a locally convex quasi C*-algebra of operators (Theorems 7.5.2, 7.5.4); (b) a functional calculus for the commutatively positive elements in A[τ ] (Theorem 7.5.7). Lemma 7.5.1 Let π be a *-representation of (A[τ ], A0 ) with domain D(π ), dense in Hπ . Let also B be an admissible subset of B(π(A)). The following hold: 1. if π is (τ, ts ∗ )-continuous, then (π(A)[ts ∗ ], π(A0 )) is a locally convex quasi C*algebra; 2. if π is (τ, tu∗ (B))-continuous, then (π(A)[tu∗ (B)], π(A0 )) is a locally convex quasi C*-algebra. Proof Clearly π(A0 ) is a C*-algebra and  π : A[τ ] → π(A)[ts ∗ ] ⊂ π(A 0 )[ts ∗ ]

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7 Locally Convex Quasi C*-Algebras and Their Structure

is a (τ, ts ∗ )-continuous *-representation of (A[τ ], A0 ), where (π(A), π(A0 )) is  u  a quasi *-algebra and (π(A 0 )[ts ∗ ], π(A0 )) (similarly (π(A0 )[t∗ (B)], π(A0 )) is a locally convex quasi C*-algebra, by Proposition 7.2.1. So, (1) and (2) follow from   Definition 7.1.3. Now, recall that a sesquilinear form ϕ on A × A is called positive, resp. invariant, if and only if, ϕ(a, a) ≥ 0, for each a ∈ A, resp. ϕ(ax, y) = ϕ(x, a ∗ y), for all a ∈ A and x, y ∈ A0 . Moreover, ϕ is called τ -continuous, if |ϕ(a, b)| ≤ p(a)p(b) for some τ -continuous seminorm p on A and all a, b ∈ A (see also Remark 2.3.1). Further, let ϕ be a τ -continuous positive invariant sesquilinear form on A0 × A0 . Then,  ϕ denotes the extension of ϕ to a τ -continuous positive invariant sesquilinear form on A × A. Moreover, let (πϕ , λϕ , Hϕ ) be the GNS construction for ϕ (see, Definition 2.4.2). Then, πϕ is extended on A, as follows: πϕ (a)λϕ (x) := lim πϕ (xα )λϕ (x), α

∀ x ∈ A0 ,

(7.5.27)

where {xα } is a net in A[τ ] with a = τ –lim xα . By the very definitions and the α

τ -continuity of ϕ, it follows that πϕ is a (τ, τs ∗ )-continuous *-representation of (A, A0 ). Now, put   P(A0 ) := τ -continuous positive invariant sesquilinear forms ϕ on A0 × A0 . We shall say that the set P(A0 ) is sufficient (see also Definition 3.1.17), whenever a ∈ A with  ϕ (a, a) = 0,

∀ ϕ ∈ P(A0 ), implies a = 0.

• From the results that follow, Theorems 7.5.2, 7.5.4 (and, of course, Corollary 7.5.3) give answers to the question (a) stated at the beginning of this section. These results can be viewed as analogues of the Gelfand–Naimark theorem, in the case of locally convex quasi C*-algebras. Theorem 7.5.2 Let (A[τ ], A0 ) be a locally convex quasi C*-algebra. The following statements are equivalent: 1. there exists a faithful, (τ, τs ∗ )-continuous *-representation π of (A, A0 ); 2. the set P(A0 ) is sufficient. Proof (1) ⇒ (2) For every ξ ∈ D(π ) define ϕξ (x, y) := π(x)ξ |π(y)ξ ,

∀ x, y ∈ A0 .

Then, {ϕξ : ξ ∈ D(π )} ⊂ P(A0 ), so that from the preceding discussion it follows easily that P(A0 ) is sufficient.

7.5 Structure of Noncommutative Locally Convex Quasi C*-Algebras

191

(2) ⇒ (1) Let ϕ ∈ P(A0 ) and (πϕ , λϕ , Hϕ ) the GNS construction for ϕ. Then, as we noticed before (see (7.5.27)), πϕ extends to a (τ, τs ∗ )-continuous *representation of (A, A0 ) with D(πϕ ) = λϕ (A0 ). Now, take  D(π ) := (λϕ (xϕ ))ϕ∈P(A0 ) ∈



Hϕ : xϕ ∈ A0 and

ϕ∈P(A0 )



λϕ (xϕ ) = 0, except for a finite number of ϕ’s from P(A0 ) and define π(a)(λϕ (xϕ )) := (λϕ (axϕ )),

∀ a ∈ A and (λϕ (xϕ )) ∈ D(π ).

Then, it is easily seen that π is a faithful, (τ, τs ∗ ) continuous *-representation of   (A, A0 ). Results for (topological) quasi *-algebras (A, A0 ), with A0 a unital C*-algebra, related to Theorem 7.5.2, have been considered in [44, Theorem 3.3] and [59, Theorem 3.2]. Now an application of Theorem 7.5.2 and Lemma 7.5.1, gives the following Corollary 7.5.3 Let (A[τ ], A0 ) be as in Theorem 7.5.2. Suppose that the set P(A0 ) is sufficient. Then, the locally convex quasi C*-algebra (A[τ ], A0 ) is continuously embedded in a locally convex quasi C*-algebra of operators. The next theorem gives further conditions under which a locally convex quasi C*-algebra A[τ ] can be continuously embedded in a locally convex quasi C*algebra of operators. Theorem 7.5.4 Let (A[τ ], A0 ) be a locally convex quasi C*-algebra. Suppose the multiplication of A0 satisfies the following condition: For every τ -bounded subset B of A0 and every λ ∈ , there exist λ ∈  and a positive constant cB , such that sup pλ (xy) ≤ cB pλ (x),

∀ x ∈ A0 .

y∈B

Then, the next statements are equivalent:

(i) there is a faithful τ, τ∗u (B) -continuous *-representation π of (A, A0 ), where B is an admissible subset of B(π(A)); (ii) there is a faithful (τ, τs ∗ )-continuous *-representation of (A, A0 ); (iii) the set P(A0 ) is sufficient. Proof (i) ⇒ (ii) is trivial (see (7.2.10)). (ii) ⇒ (iii) follows from Theorem 7.5.2.

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7 Locally Convex Quasi C*-Algebras and Their Structure

(iii) ⇒ (i) Let ϕ ∈ P(A0 ) and (πϕ , λϕ , Hϕ ) be the GNS construction for ϕ (see discussion before Theorem 7.5.2). Set   Bϕ := λϕ (B) : B a τ -bounded subset of A0 . Then, for each τ -bounded subset B of A0 , we have sup πϕ (a)λϕ (y) = sup ϕ(ay, ay)1/2 ≤ sup pλ (ay) ≤ cB pλ (a), y∈B

y∈B

y∈B

for all a ∈ A and some λ, λ ∈ .

It is clear now that λϕ (B) ∈ B(πϕ (A)) and that (see (7.2.9)) πϕ is τ, τ∗u (Bϕ ) -continuous. Let now π be as in the proof of Theorem 7.5.2. Put  Bπ :=

finite 

λϕ (Bϕ ) : Bϕ

 a τ -bounded subset of A0 .

ϕ∈P(A0 )

seen that Bπ is an admissible subset of B(π(A)) and π a faithful, Then,u it is easily

τ, τ∗ (Bπ ) -continuous *-representation of A.   An analogue of Corollary 7.5.3 is stated in the case of Theorem 7.5.4, too. Taking again (A[τ ], A0 ) as in Theorem 7.5.2, we proceed to the study of a functional calculus for the commutatively positive elements of A[τ ] (see (b) at the beginning of this section). So, let a ∈ A+ c (for this notation see discussion before Proposition 7.1.2). Then, from Proposition 7.1.2(1), the element (e + a)−1 exists ∗ and belongs to U (A+ 0 ). Consider the maximal commutative C*-subalgebra C (a) of −1 A0 containing the elements e, (e + a) . Then, • C ∗ (a)[τ ] satisfies the properties (T1 )–(T4 ) of Sect. 7.1. The properties

(T1 )–(T3 ) are trivially checked. To check (T4 ), we must prove that U C ∗ (a)+ is τ -closed.

So, let {xα } be a net in U C ∗ (a)+ , such that xα − → x. But, U C ∗ (a)+ ⊂ U (A+ 0 ) τ

+ and since U (A+ 0 ) is τ -closed we have that x ∈ U (A0 ). On the other hand,

→ yx, xy ← − xα y = yxα − τ

τ

∀ y ∈ C ∗ (a).

∗ ∗ Hence, xy = yx, which

by the maximality of C (a) means that x ∈ C (a) and ∗ + finally x ∈ U C (a) . This completes the proof of (T4 ) and all the above lead to the following

Proposition 7.5.5 Let (A[τ ], A0 ) be a locally convex quasi C*-algebra. A+ c  Let a ∈−1 ∗ and C (a) the maximal commutative C*-subalgebra of A0 containing e, (e+a) .

∗ (a)[τ ], C ∗ (a) is a commutative locally convex quasi C*-algebra.  Then, C ∗ (a)[τ ]+ .  Corollary 7.5.6 The element a, as before, belongs to C

7.5 Structure of Noncommutative Locally Convex Quasi C*-Algebras

193

Proof Since a ∈ A+ c , Proposition 7.1.2(2) implies that a(e + εa)−1 =

 1 e − (e + εa)−1 ∈ A+ 0 , ε

∀ ε > 0.

Now, (e + a)−1 commutes with every element ω ∈ C ∗ (a), therefore ω also commutes with e + a, hence with a, consequently with (e + εa)−1 too. Thus, a(e + εa)−1 ∈ C ∗ (a), for each ε > 0. Since moreover, a = τ – lim a(e + εa)−1 ε→0

∗ (a)[τ ]+ .  (ibid.), Definition 7.1.1 gives that a ∈ C

 

It is now clear from Corollary 7.5.6 that making use of Theorem 7.4.6 for ∗  C (a)[τ ]+ , we can obtain the promised functional calculus for the commutatively positive elements of the noncommutative locally convex quasi C*-algebra (A[τ ], A0 ). That is, we have the following Theorem 7.5.7 Let (A[τ ], A0 ) be a noncommutative locally convex quasi C*n algebra. Let a ∈ A+ c such that a is well defined for some n ∈ N. Then, there is n

, a unique *-isomorphism f → f (a) from Ck σC ∗ (a) (a) into A[τ ], such that k=1

1. if u0 ∈

n ,



Ck σC ∗ (a) (a) with u0 (λ) = 1, for each λ ∈ σC ∗ (a) (a), then u0 (a) =

k=1

e ∈ C ∗ (a) → A[τ ]; n

, 2. if u1 ∈ Ck σC ∗ (a) (a) with u1 (λ) = λ, for each λ ∈ σC ∗ (a) (a), then u1 (a) = k=1

a ∈ A[τ ]; n

, Ck σC ∗ (a) (a) and ϕ ∈ M(C ∗ (a)); 3. f(a)(ϕ) = f ( a (ϕ)), for any f ∈ k=1

4. (f1 + f2 )(a) = f1 (a) + f2 (a), for any f1 , f2 ∈

n ,

Ck (σC ∗ (a) (a)), (λf )(a) =

k=1

n

, λf (a), for any f ∈ Ck σC ∗ (a) (a) and λ ∈ C, (f1 f2 )(a) = f1 (a)f2 (a), for k=1 any fj ∈ Ckj σC ∗ (a) (a) , j = 1, 2, with k1 + k2 ≤ n;

5. restricted to Cb σC ∗ (a) (a) the map f → f (a) is an isometric *-isomorphism of the C*-algebra Cb σC ∗ (a) (a) onto the closed *-subalgebra of the C*-algebra C ∗ (a) generated by e and (e + a)−1 .

Now, an application of Corollary 7.4.7 for the commutative locally convex quasi

∗ (a)[τ ], C ∗ (a) and Theorem 7.5.7 give the following  C*-algebra C Corollary 7.5.8 Let (A[τ ], A0 ) be as in Theorem 7.5.7. Let a ∈ A+ c and n ∈ N. n . The element b is called , such that a = b Then, there is a unique element b ∈ A+ c 1 1 commutatively nth-root of a and is denoted by a n . If n = 2, the element a 2 is called commutatively square-root of a.

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7 Locally Convex Quasi C*-Algebras and Their Structure

7.6 Locally Convex Quasi C*-Algebras and Noncommutative Integration In the case of Banach quasi *-algebras an important role has been played by the two sets of sesquilinear forms SA0 (A) and TA0 (A) (see Sects. 3.1.2 and 5.6.4). We extend these notions to the more general set-up of locally convex quasi *-algebras in a natural way. Definition 7.6.1 Let (A[τ ], A0 ) be a locally convex quasi C*-algebra with unit τ e. We denote by SA (A) the set of all sesquilinear forms ϕ ∈ QA0 (A) with the 0 following additional properties: (i) |ϕ(a, b)| ≤ p(a)p(b), for some τ -continuous seminorm p on A and all a, b ∈ A; (ii) ϕ(e, e) ≤ 1. The locally convex quasi C*-algebra (A[τ ], A0 ) is called *-semisimple if a ∈ A, τ ϕ(a, a) = 0, for every ϕ ∈ SA (A), implies a = 0. 0 τ τ We denote by TA0 (A) the set of all sesquilinear forms ϕ from SA (A), with the 0 following property (iii) ϕ(a, a) = ϕ(a ∗ , a ∗ ),

∀ a ∈ A.

Remark 7.6.2 Notice that • By (iii) of Definition 7.6.1 and by polarization, we get ϕ(a, b) = ϕ(b∗ , a ∗ ),

∀ a, b ∈ A.

• The set TAτ 0 (A) is convex. Let M be a von Neumann algebra on a Hilbert space H and  a normal faithful semifinite trace on M+ , then, as shown in Proposition 5.6.4, (Lp (), L∞ () ∩ Lp ()) is a Banach quasi *-algebra and if  is a finite trace, (Lp (), M) is a CQ*algebra. Example 7.6.3 Let M be a von Neumann algebra and  a normal faithful semifinite trace on M+ . Then, (see Sect. 5.6.2), (Lp (), Jp ), p ≥ 2, is a *-semisimple Banach quasi *-algebra. If  is a finite trace (we assume (e) = 1), then (Lp (), M), with p ≥ 2, is a *-semisimple locally convex quasi C*-algebra. If p ≥ 2, Lp spaces possess a sufficient family of positive sesquilinear forms. Indeed, in this case, since for every W ∈ Lp (), |W |p−2 ∈ Lp/(p−2) (), then the sesquilinear form ϕW defined by ϕW (A, B) :=

 A(B|W |p−2 )∗ p−2 W p,

,

A, B ∈ Lp ()

7.6 Locally Convex Quasi C*-Algebras and Noncommutative Integration

195

satisfies the conditions of Definition 7.6.1, (see [52], for more details). Moreover, p

ϕW (W, W ) = W p, . Definition 7.6.4 Let M be a von Neumann algebra and  a normal faithful semifinite trace defined on M+ . We say that a measurable operator T belongs to p Lloc () if T P ∈ Lp (), for every central -finite projection P of M. p

Remark 7.6.5 The von Neumann algebra M is a subset of Lloc (). Indeed, if a ∈ M, then for every -finite central projection P of M the product XP belongs to the *-ideal Jp (see discussion at the beginning of Sect. 5.6.2). Throughout this section   we are given a von Neumann algebra M on a Hilbert space H with a family Pj j ∈J of -finite central projections of M, such that Pm = 0 (i.e., the Pj ’s are orthogonal); • if #l, m ∈ J , l = m, then Pl# • j ∈J Pj =  e; where j ∈J Pj denotes the projection onto the subspace  generated by Pj H j ∈J . The previous two conditions are always realized in a von Neumann algebra M with a faithful normal semifinite trace (see Lemma 5.6.1 and [15, 27], for more details). If  is a normal faithful semifinite trace on M+ , we define, for each a ∈ M, the following seminorms qj (a) := XPj p, , j ∈ J ,on M. The translation-invariant locally convex topology defined by the system qj j ∈J is denoted by τp . Definition 7.6.6 Let M be a von Neumann algebra and  a normal faithful  τp the τp -completion of M. semifinite trace defined on M+ . We denote by M Proposition 7.6.7 Let M be a von Neumann algebra and  a normal faithful p  τp . Moreover, if there exists a family semifinite trace on M+ . Then, Lloc () ⊆ M   p  τp . Pj j ∈J as above, where all Pj ’ s are mutually equivalent, then Lloc () = M p

p

Proof From Remark 7.6.5, M ⊆ Lloc (). If Y ∈ Lloc (), for every j ∈ J , we have  j ∞ Y Pj ∈ Lp (). Then, for every j ∈ J , there exists a sequence Xn n=1 ⊆ Jp , such j

that Xn − Y Pj p, −→ 0. n→∞

Let FJ be the family of finite subsets of J ordered by inclusion and F ∈ FJ . We put Tn,F :=



j

Xn Pj ∈ M.

j ∈F

Then, the net {Tn,F } converges to Y with respect to τp . Indeed, for every m ∈ J , qm (Tn,F − Y ) = (Tn,F − Y )Pm p, = (Xnm − Y )Pm p, ,

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7 Locally Convex Quasi C*-Algebras and Their Structure

for sufficiently large F . Thus, the inequality (Xnm − Y )Pm p, ≤ Xnm − Y Pm p, implies that qm (Tn,F − Y ) −→ 0. n,F

p  τp . On the other hand, assume that all Pj ’s are mutually Hence, Lloc () ⊆ M    τp , there exists a net Xα ⊆ M, such that Xα → Y , equivalent. Then, if Y ∈ M with respect to τp ; consequently,

Xα Pj → Y Pj ∈ Lp (), with respect to  · p, .

(7.6.28)

But, for each central -finite projection P , we have    

Pj =  P Pj . (P ) =  P j ∈J

(7.6.29)

j ∈J

By our assumption, for any l, m ∈ J , we may pick U ∈ M, such that U ∗ U = Pl and U U ∗ = Pm , hence, (P Pl ) = (P U ∗ U ) = (U P U ∗ ) = (P U U ∗ ) = (P Pm ). So, all terms on the right hand side of (7.6.29) are equal and since the above series converges, only a finite number of them can be nonzero. Thus, for some s ∈ N we may write J = {1, . . . , s} and then P =P



Pj = P

j ∈J

s  j =1

Pj =

s 

P Pj

(7.6.30)

j =1

and thus YP =

s  j =1

Y P Pj =

s 

Y Pj P ∈ Lp ().

(7.6.31)

j =1

 τp , for each central -finite projection P , we have Y P ∈ Lp (). Therefore, if Y ∈ M p  τp . Hence, Lloc () ⊇ M   Remark 7.6.8 In general,  it is not guaranteed that a von Neumann algebra possesses an orthogonal family Pj j ∈J of mutually equivalent finite central projections, such # p  τp . that j ∈J Pj = e; but, if this is the case, then Lloc () = M Theorem 7.6.9 Let M be a von Neumann algebra on a Hilbert space H and  a  τp , M) is a locally convex quasi normal, faithful, semifinite trace on M+ . Then, (M C*-algebra, with respect to τp ,, consisting of measurable operators.

7.7 The Representation Theorems

197

Proof The topology τp satisfies the properties (T1 )–(T4 ). We just prove here (T3 ) and (T4 ). • (T3 ) For each λ ∈ J , qλ (XY ) = Pλ XY p, ≤ XPλ Y p, = Xqλ (Y ),

∀ X, Y ∈ M;

  • (T4 ) The set U (M+ ) := X ∈ M+ : X ≤ 1 is τp -closed. To see this consider a net {Fα } in U (M+ ), such that Fα → F , with respect to the topology τp . Then, for each j ∈ J , we have (Fα − F )Pj p, → 0. By assumption on Pj , the trace  is a normal faithful finite trace on the von Neumann algebra (Pj M)+ and by Proposition 5.6.4 (Lp (), Pj M) is a CQ*-algebra. Therefore, using (T4 ) for (Lp (), Pj M), we have F Pj ∈ U ((Pj M)+ ), for each j ∈ J . This, .by definition, implies that F ∈ M. Indeed, for every h = j ∈J Pj h ∈ H = j ∈J Pj H, we have    F Pj h2 = F Pj Pj h2 ≤ Pj h2 = h2 . F h2 = j ∈J

j ∈J

j ∈J

Hence, F ∈ U (M+ ).

 

Remark 7.6.10 By Proposition 7.6.7, C*-algebra, with respect to τp .

p (Lloc (), M)

itself is a locally convex quasi

7.7 The Representation Theorems The results of this section generalize to locally convex quasi C*-algebras those obtained in Sect. 5.6.4 in the case of CQ*-algebras. Let (A[τ ], A0 ) be a locally convex quasi C*-algebra with a unit e. For each ϕ ∈ TAτ 0 (A), we define a linear functional ωϕ on A0 by ωϕ (x) := ϕ(x, e),

∀ x ∈ A0 .

Then, we have ωϕ (x ∗ x) = ϕ(x ∗ x, e) = ϕ(x, x) = ϕ(x ∗ , x ∗ ) = ωϕ (xx ∗ ) ≥ 0 and this shows at once that ωϕ is positive and tracial. We denote by π the universal *-representation of A0 (Remark A.6.16) and for every ϕ ∈ TAτ 0 (A) and x ∈ A0 , we put ρϕ (π(x)) := ωϕ (x).

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7 Locally Convex Quasi C*-Algebras and Their Structure

Then, for each ϕ ∈ TAτ 0 (A), ρϕ is a positive, bounded, linear functional on the operator algebra π(A0 ). Clearly, ρϕ (π(x)) = ωϕ (x) = ϕ(x, e),

∀ x ∈ A0 .

If {pλ } is a directed family of seminorms defining the topology of A, by the fact that {pλ } is directed, we conclude that there exist γ > 0 and λ ∈ , such that | ρϕ (π(x)) | = | ωϕ (x) | = | ϕ(x, e) | ≤ γ 2 pλ (xe)pλ (e),

∀ x ∈ A0 .

Then, combining also with (T3 ), we obtain | ρϕ (π(x)) | ≤ γ 2 x0 pλ (e)2 , for some λ ∈  and ∀ x ∈ A0 . Thus, ρϕ is continuous on π(A0 ). By [64, Vol. 2, Proposition 10.1.1], ρϕ is weakly continuous and so it extends  uniquely to π(A0 ) , by the Hahn–Banach theorem. Moreover, since ρϕ is a trace on  π(A0 ), the extension ρϕ is also a trace on the von Neumann algebra M := π(A0 ) generated by π(A0 ).   Clearly, the set NTAτ (A) ≡ ρϕ : ϕ ∈ TAτ 0 (A) is convex. 0

Definition 7.7.1 The locally convex quasi C*-algebra (A[τ ], A0 ) is called strongly *-semisimple if the following conditions are fulfilled: (a) x ∈ A0 and ϕ(x, x) = 0, for every ϕ ∈ TAτ 0 (A), imply x = 0; (b) the set NTAτ (A) (see also Proposition 3.6.14 and discussion before it) is w*0 closed. Note that if (A[τ ], A0 ) is a CQ*-algebra, (b) is automatically satisfied, according to Proposition 5.6.14. Example 7.7.2 Let M be a von Neumann algebra and  a normal, faithful, semifinite trace on M+ . Then, as seen in Example 7.6.3, if  is a finite trace, (Lp (), M), with p ≥ 2, is a *-semisimple, locally convex quasi C*-algebra. Moreover, the conditions (a) and (b) of Definition 7.7.1 are satisfied. Indeed, in this case, the set NTAτ (A) is w*-closed by Proposition 5.6.14. Therefore, (Lp (), M), 0 with  finite, is a strongly *-semisimple, locally convex quasi C*-algebra. Let (A[τ ], A0 ) be a locally convex quasi C*-algebra with unit e, π the universal  representation of A0 and M = π(A0 ) . Recall that f  denotes the norm of a bounded functional f on M and M the topological dual of M; then, the norm ρϕ  of the linear functional ρϕ on M, equals to the norm ρϕ  of the linear ϕ (π(e)) = ϕ(e, e) functional ρϕ on π(A0 ). By (ii) of Definition 7.6.1, ρϕ  = ρ ≤ 1. Hence, if (b) of Definition 7.7.1 is satisfied, the set NTAτ (A) , being a w*-closed 0 subset of the unit ball of M , is w*-compact.

7.7 The Representation Theorems

199

Let ENTAτ (A) be the set of extreme points of NTAτ (A) ; then, NTAτ (A) coincides 0 0 0 with the w*-closure of the convex hull of ENTAτ (A) . 0 Moreover, ENTAτ (A) is a family of normal finite traces on the von Neumann 0 algebra M.   We put F := ϕ ∈ TAτ 0 (A) : ρϕ ∈ ENTAτ (A) and denote by Pϕ , the support 0   projection, corresponding to the trace ρϕ . By [Lemma 5.6.12], Pϕ ϕ∈F consists of # mutually orthogonal projections and if Q := ϕ∈F Pϕ , then μ :=



ρϕ

ρ  ϕ ∈ENT τ (A) A0

is a normal, faithful, semifinite trace defined on the direct sum (see [27] and [89])  QM ≡ Pϕ M. ϕ∈F

of von Neumann algebras. For the topology τ2 used in the following theorem, see discussion before Definition 7.6.6. Theorem 7.7.3 Let (A[τ ], A0 ) be a strongly *-semisimple locally convex quasi C*algebra with unit e and π the universal representation of A0 . Then, there exists a monomorphism * τ2 ,  : a ∈ A → (a) :=  a ∈ QM with the following properties: (i)  extends the isometry π : A0 → B(H) given by the Gelfand–Naimark theorem; (ii) (a ∗ ) = (a)∗ , for every a ∈ A; (iii) (ab) = (a)(b), for all a, b ∈ A, such that either a ∈ A0 or b ∈ A0 . Proof Let {pλ }λ∈ be, as before, the family of seminorms defining the topology τ of A. Let a ∈ A be fixed. Then, there exists a net {xα }α∈ of elements of A0 , such that pλ (xα − a) → 0, for each λ ∈ . We put aα = π(xα ). By (i) of Definition 7.6.1, for every ϕ ∈ TAτ 0 (A), there exist γ > 0 and λ ∈ , such that, for each α, β ∈ ,

Pϕ (aα − aβ )2, ρϕ = Pϕ π(xα ) − π(xβ ) 2, ρϕ

1/2 = ρϕ (|Pϕ (π(xα ) − π(xβ ))|2 ) =

1/2 = ϕ (xα − xβ )∗ (xα − xβ ), e

1/2 = ϕ(xα − xβ , xα − xβ ) ≤ γ pλ (xα − xβ ) −→ 0. α,β

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2  ). Clearly, a = P a . Let aϕ be the  · 2, ϕϕ -limit of the net {Pϕ aα } in L (ρ ϕ ϕ ϕ ϕ We define  Pϕ aϕ =:  a. (a) := ϕ∈F

* τ2 . It is easy to see that the map a # A →  * τ2 is well Evidently,  a ∈ QM a ∈ QM defined and injective. Indeed, if xα → 0, there exist γ > 0 and λ ∈ , such that Pϕ aα 2, ρϕ = Pϕ π(xα )2, ρϕ

1/2 = ρϕ (|Pϕ (π(xα )|2 ) = ∗

1/2 = ϕ xα xα , e

1/2 = ϕ(xα , xα ) ≤ γ pλ (xα ) → 0, 2  ). Thus, P (a ) = 0, for every where  · 2, ρϕ clearly denotes the norm in L (ρ ϕ ϕ α τ ϕ ∈ TA0 (A), therefore  a = 0. Moreover, if Pϕ a = 0, for each ϕ ∈ F, then ϕ(a, a) = 0, for every ϕ ∈ F. Since, every ϕ ∈ TAτ 0 (A) is a w*-limit of convex combinations of elements from F, we obtain ϕ(a, a) = 0, for every ϕ ∈ TAτ 0 (A). Hence, by assumption, a = 0.  

Remark 7.7.4 In the same way one proves that • If (A[τ ], A0 ) is a strongly *-semisimple, locally convex quasi C*-algebra and there exists a faithful ϕ ∈ TAτ 0 (A) (i.e., the equality ϕ(a, a) = 0, implies a = 0), then there exists a monomorphism  : a ∈ A → (a) :=  a ∈ L2 (ρϕ ), with the following properties: (i)  extends the isometry π : A0 → B(H) given by the Gelfand–Naimark theorem; (ii) (a ∗ ) = (a)∗ , for every a ∈ A; (iii) (ab) = (a)(b), for all a, b ∈ A, such that either a ∈ A0 or b ∈ A0 .  • If the semifinite von Neumann algebra   π(A0 ) admitsan orthogonal family of mutually equivalent projections P i i∈I , such that i∈I P i = e, then it is easy to see that the map a ∈ A →  a ∈ L2loc () (cf. Definition 7.6.4), is a monomorphism.

Appendix A

*-Algebras and Representations

This chapter is devoted to general aspects of the theory of *-algebras and their representations.

A.1 Algebras: Basic Definitions A vector space A is said to be an algebra if a map (a, b) ∈ A × A → ab ∈ A is defined and satisfies the following conditions: (i) a(bc) = (ab)c, (ii) (a + b)c = ac + bc and a(b + c) = ab + ac, (iii) α(ab) = (αa)b = a(αb), for all a, b, c ∈ A and α ∈ C. The element ab of A is called the product of a and b. An element e of A is called a unit of A if ea = ae = a, for each a ∈ A; this element, if it exists, is necessarily unique. An algebra A is said to be a *-algebra if there exists a conjugate linear map a ∈ A → a ∗ ∈ A such that (ab)∗ = b∗ a ∗ and (a ∗ )∗ = a for all a, b ∈ A. The map a ∈ A → a ∗ ∈ A is called an involution of A. If A has a unit e, then e∗ = e. Remark A.1.1 If the *-algebra A has no unit, there is a standard procedure for embedding it in a *-algebra with unit Ae called unitization of A. Indeed, let us consider the space Ae := A⊕C. For (a, λ), (b, μ) ∈ Ae and α ∈ C, define algebraic operations and involution as follows: (a, λ) + (b, μ) := (a + b, λ + μ),

α(a, λ) := (αa, αλ);

(a, λ) · (b, μ) := (ab + μa + λb, λμ),

(a, λ)∗ := (a ∗ , λ).

© Springer Nature Switzerland AG 2020 M. Fragoulopoulou, C. Trapani, Locally Convex Quasi *-Algebras and their Representations, Lecture Notes in Mathematics 2257, https://doi.org/10.1007/978-3-030-37705-2

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Then, it is easily seen that, with these operations and involution, Ae is a *-algebra with unit (0, 1), such that a ∈ A → (a, 0) ∈ Ae , for all a ∈ A. • From now on, for convenience, we put e ≡ (0, 1) in Ae . Example A.1.2 Very familiar examples of *-algebras are provided by certain spaces of complex valued functions. In all cases, the involution is defined by complex conjugation. Take, for instance, the space C(X) of all continuous functions on a locally compact Hausdorff space X or the space Cc∞ (R) of all C-valued C ∞ functions on R, with compact support. Example A.1.3 Let D be a dense subspace of a Hilbert space H. We denote with L† (D) the space of all closable operators A in H, such that D(A) = D, D(A∗ ) ⊃ D (where D(A) means domain of A, as we have noticed at the beginning of Sect. 2.1.3) and both A and A∗ map D into itself. In this case, we define an involution A → A† on L† (D) by putting A† := A∗ D . It is easy to verify that with this involution and the natural algebraic operations L† (D) is a *-algebra. A normed algebra A is an algebra, which is also a normed space with norm  · , such that ab ≤ ab,

∀ a, b ∈ A.

If A is a Banach space under the norm  · , we call it a Banach algebra. If A has a unit e, we shall always suppose, without loss of generality, that e = 1. This follows from a theorem of Gelfand, according to which one can define a second norm  ·  on A making it a Banach algebra and such that  ·  is equivalent to  ·  with e = 1. A normed *-algebra, (respectively, Banach *-algebra) is a *-algebra A, which is also a normed (respectively, Banach algebra) with norm  · , such that a ∗  = a,

∀ a ∈ A.

An involution with the previous property is called isometric involution. A Banach *-algebra A is said to be a C*-algebra if the given norm  ·  satisfies the so-called C*-condition, namely a ∗ a = a2 ,

∀ a ∈ A.

Example A.1.4 The space C(X) of C-valued continuous functions on a compact Hausdorff space X provides the simplest commutative example of a C*-algebra with unit. The norm is defined, as usual, by f ∞ := sup |f (x)|,

∀ f ∈ C(X).

x∈X

It is easy to see that this norm satisfies the C*-condition.

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Example A.1.5 The space B(H) of all bounded linear operators on a Hilbert space H is a *-algebra under the natural algebraic operations and the map A → A∗ as involution. As is already seen in Chap. 2, with the operator norm   A := sup Aξ  : ξ ∈ H, ξ  = 1 ,

A ∈ B(H),

(A.1.1)

B(H) is a normed *-algebra with isometric involution. We now prove that B(H) is complete under the previous norm (A.1.1), thus it will be a Banach *-algebra. So let {An } be a Cauchy sequence in B(H). For each ξ ∈ H, the sequence {An ξ } is a Cauchy sequence in H. Then, it converges to a vector ξ ∗ ∈ H. We put Aξ = ξ ∗ . It is easily checked that A is well-defined and linear. To see that A is bounded, we take into account that the sequence {An } is bounded (i.e., there exists M > 0, such that An  ≤ M, for every n ∈ N); thus Aξ  = lim An ξ  ≤ Mξ . n→∞

It remains to prove that An → A, with respect to the operator norm. Since {An } is Cauchy, for every > 0, there exists n , such that for n, m > n , An − Am  < . Now fix n > n . Then, we have (An − A)ξ  = lim (An − Am )ξ  ≤ lim An − Am ξ  ≤ ξ . m→∞

m→∞

Hence, An − A = sup (An − A)ξ  ≤ , ξ =1

which proves the claim. We now show that B(H) is actually a (non-commutative) C*-algebra, i.e., that the C*-condition is fulfilled. First we have A∗ A ≤ A2 , for all A ∈ B(H). On the other hand, A∗ A = =

sup

|A∗ Aξ |η|

sup

|Aξ |Aη|

ξ =η=1 ξ =η=1

≥ sup |Aξ |Aξ | = A2 , ξ =1

∀ A ∈ B(H).

Example A.1.6 An operator B ∈ B(H) is called compact if B maps bounded sets of H into relatively compact subsets of H, or equivalently, if for any bounded sequence {ξn } in H, the sequence {Bξn } contains a convergent subsequence. Let C(H) denote the space of all compact operators on H. Then, C(H) is a closed two-sided ideal of B(H) and it is itself a C*-algebra. The quotient algebra B(H)/C(H) is a Banach

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*-algebra under the quotient norm   [A]C := inf A + B : B ∈ C(H) , where [A] denotes the equivalent class of A. This Banach *-algebra is called Calkin algebra. Example A.1.7 Let M be a subset of B(H). The (von Neumann) commutant of M is defined as M = {Y ∈ B(H) : XY = Y X, ∀X ∈ M}. If M is *-invariant (i.e., X ∈ M ⇔ X∗ ∈ M), then M is a *-algebra. The double commutant (called also bicommutant) M is defined as M := (M ) . If M = M , M is said to be a von Neumann algebra. Due to von Neumann double commutant theorem [27, vol. I, Section 2.3], every von Neumann algebra is norm-closed in B(H) and so it is a C*-algebra.

A.2 Representations Definition A.2.1 Let A be an algebra. A representation of A on a vector space Dπ is a homomorphism π of A into the algebra L(Dπ ) of all linear operators on Dπ ; that is (i) π(a + b) = π(a) + π(b), ∀ a, b ∈ A; (ii) π(αa) = απ(a), ∀ a ∈ A, α ∈ C; (iii) π(ab) = π(a)π(b), ∀ a, b ∈ A. A representation π ona vector space Dπ is called ultracyclic, if there exists ξ0 ∈ Dπ , such that Dπ = π(a)ξ0 : a ∈ A . A subspace M of Dπ is said to be invariant under π if π(a)M ⊆ M, for every a ∈ A. A representation π on a vector space Dπ is called algebraically irreducible, if only the trivial subspaces {0} and Dπ are invariant subspaces of Dπ , under π . A representation π of A in a normed space Dπ is called bounded if the operator π(a) is bounded, for every a ∈ A. If A is a *-algebra and Dπ a pre-Hilbert space, a *-representation of A on Dπ is a *-homomorphism of A into L† (Dπ ), that is a homomorphism satisfying, in addition to the properties listed in Definition A.2.1, the property π(a ∗ ) = π(a)† , for every a ∈ A. Let Hπ denote the Hilbert space completion of Dπ . If Dπ1 is another dense subspace of Hπ with Dπ ⊆ Dπ1 and π1 is a *-representation of A defined on Dπ1 , then π is said to be a sub *-representation of π1 , writing π ⊆ π1 , if π1 (a)ξ = π(a)ξ , for any a ∈ A and ξ ∈ Dπ .

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Remark A.2.2 Let π be a *-representation of a *-algebra A without unit, defined on Dπ . Then, π can be extended to a *-representation π e of the unitization Ae of A (see Remark A.1.1) by putting

π e (a, λ) = π(a) + λI,

∀ (a, λ) ∈ Ae ,

where I denotes the identity operator of Dπ . It is easily seen that π e is a *-representation on the same domain Dπ , with the property π e e ≡ (0, 1) = I. Remark A.2.3 If π is a *-representation of A on Dπ , then Dπ can be endowed with a topology, denoted by tπ , linked to the representation π itself. More precisely, tπ is the locally convex topology on Dπ defined, for a given a ∈ A, by the family of seminorms ξ → ξ π(a) := ξ  + π(a)ξ ,

∀ ξ ∈ Dπ ,

(A.2.1)

where  ·  is the Hilbert norm. The topology tπ is known as the graph topology on Dπ . Let π be a *-representation of the *-algebra A on the pre-Hilbert space Dπ . We put D π =



D(π(a)),

a∈A

where π(a) denotes the closure of the operator π(a) (see beginning of Sect. 2.1.3). Then, we define  π (a) := π(a) Dπ ,

a ∈ A.

Lemma A.2.4 Let {a1 , . . . , an } be a finite subset of A and π a *-representation of A on Dπ . Then, there exists b ∈ A, such that n 

π(ai )ξ  ≤ π(b)ξ ,

∀ ξ ∈ Dπ .

i=1

 Proof First add a unit, if necessary. Then, put b = e + n2 ni=1 ai∗ ai . If b1 =   n n n n ∗ ∗ i=1 ai ai , we obtain π(b1 )ξ |ξ  = 2 i=1 π(ai ai )ξ |ξ , hence 2 π(b)ξ 2 = ξ 2 + 2π(b1 )ξ |ξ  + π(b1 )ξ 2 ≥ n

=n

n  i=1

π(ai )ξ 2 ≥

 n  i=1

i=1

2 π(ai )ξ 

n  π(ai∗ ai )ξ |ξ 

.  

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The latter inequality is nothing but the Cauchy–Schwarz inequality. Theorem A.2.5 For any *-representation π of A on Dπ ,  π is a *-representation of A on D . π Proof We may suppose, without loss of generality, that A has a unit e. We show first that  π (b)D π ⊂ D π , for all b ∈ A. Let a, b ∈ A. From Lemma A.2.4, there exists c ∈ A, such that π(c)ξ  ≥ π(ab)ξ  + π(b)ξ  + ξ ,

∀ ξ ∈ D π.

(A.2.2)

If ξ ∈ D π , then ξ ∈ D(π(c)); thus, there exists a sequence {ξn } ⊂ Dπ , such that ξn → ξ and π(c)ξn → π(c)ξ . From (A.2.2) the following hold 

π(b)(ξn − ξm ) → 0, π(a)π(b)(ξn − ξm ) = π(ab)(ξn − ξm ) → 0,

that imply ξ ∈ D(π(b)), π(b)ξn → π(b)ξ and π(b)ξ ∈ D(π(a)). Since a ∈ A is arbitrary, it follows that π(b)D π ⊂ D π . Moreover, π(ab)ξ = π(a)π(b)ξ . Therefore,  π is a representation of A in D π . To prove that it is a *-representation, we take ξ, η ∈ D π . Then, there exist two sequences {ξn }, {ηn } in D(π ), such that ξn → ξ , ηn → η and π(a)ξn → π(a)ξ, π(a ∗ )ηn → π(a ∗ )η. Hence, π (a ∗ )η  π (a)ξ |η = lim π(a)ξn |ηn  = lim ξn |π(a ∗ )ηn  = ξ | n→∞

and this concludes the proof.

n→∞

 

The *-representation  π is called the closure of π (see discussion after Definiπ = π and π is called closed (ibid.). More generally, tion 2.2.5). If D π = Dπ , then  it can be proved that  π is the minimal closed extension of π and this fact motivates its name. Remark A.2.6 From the construction itself of D π it follows that Dπ is dense in D π with respect to the locally convex topology defined by (A.2.1). Moreover, it can be proved that D π can be identified with the completion of Dπ in this topology. If π is a bounded *-representation, then for each a ∈ A,  π (a) is an everywhere defined bounded operator on the Hilbert space Hπ , completion of Dπ , with respect to the norm defined by the inner product. More precisely, we have Theorem A.2.7 Let π be a *-representation of A on Dπ ⊂ Hπ . The following statements are equivalent: (i) π is bounded on Dπ ; (ii) π is the restriction to Dπ of a bounded *-representation π1 on A, with Dπ1 = Hπ ; (iii) D π is bounded. π = Hπ and 

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Proof (i) ⇒ (ii) Let π be bounded. Then, for every a ∈ A, the operator π(a) is bounded and has a bounded everywhere defined closure π(a) on Hπ . Define π1 (a) := π(a),

a ∈ A.

Then, π1 is a *-representation of A with domain Dπ1 = Hπ and clearly π ⊆ π1 . (ii) ⇒ (iii) Since π ⊆ π1 , with π1 bounded, π itself is bounded and so π(a) is a bounded everywhere defined operator on Hπ , for every a ∈ A. Hence, D π =



D(π(a)) = Hπ

a∈A

and  π is bounded. (iii) ⇒ (i) This is obvious.

 

Definition A.2.8 Let A be a *-algebra, Hπ , Hρ two Hilbert spaces, π and ρ two *-representations of A with domains Dπ ⊆ Hπ and Dρ ⊆ Hρ , respectively. We say that π and ρ are unitarily equivalent (and we write π ≈ ρ) if there exists a unitary operator U : Hπ → Hρ , such that U Dπ = Dρ and ρ(a)U ξ = U π(a)ξ,

∀ a ∈ A, ξ ∈ Dπ .

Lemma A.2.9 Let π and ρ be two *-representations of A with domains Dπ ⊆ Hπ and Dρ ⊆ Hρ , respectively. If π ≈ ρ, then  π ≈ρ . Proof Let U : Hπ → Hρ be a unitary operator satisfying the two conditions of Definition A.2.8. Let ξ ∈ D(π(a)), a ∈ A. Then, there exists a sequence {ξn } of elements of Dπ , such that ξn → ξ and π(a)ξn → π(a)ξ . Moreover, U ξn → U ξ and ρ(U ξn − U ξm ) = U π(a)(ξn − ξm ) = π(a)(ξn − ξm ) → 0, as n, m → ∞. Hence, U ξ ∈ D(ρ(a)) and ρ(a)U ξ = lim ρ(a)U ξn = lim U π(a)ξn = U π(a)ξ. n→∞

n→∞

(A.2.3)

From the arbitrariness of a ∈ A, it follows that U D ρ ); interchanging the π ⊂ D( roles of π and ρ and using U −1 instead of U , one can prove the converse inclusion. Finally, by (A.2.3), one easily obtains ρ (a)U ξ = U π (a)ξ,

∀ a ∈ A, ξ ∈ D π.

 

The definition of an ultracyclic representation given at the beginning of this section is of purely algebraic nature. In what follows, one takes into account the topological structure of the domains involved.

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Definition A.2.10 A *-representation π of a *-algebra A with domain Dπ is called cyclic if there exists ξ0 ∈ Dπ , such that {π(a)ξ0 : a ∈ A} is norm-dense in Hπ ; it is called strongly-cyclic if there exists ξ0 ∈ Dπ , such that {π(a)ξ0 : a ∈ A} is tπ -dense in D π. The vector ξ0 is called cyclic, or strongly-cyclic for π , respectively. Theorem A.2.11 Let π and ρ be two closed *-representations of A with domains Dπ ⊆ Hπ and Dρ ⊆ Hρ , respectively. Assume that π is cyclic (respectively, strongly-cyclic), with cyclic (respectively, strongly-cyclic) vector ξ0 . Then, ρ is unitarily equivalent to π , if and only if, ρ is cyclic (respectively, strongly-cyclic) and there exists a cyclic (respectively, strongly-cyclic) vector η0 ∈ Dρ , such that π(a)ξ0 |ξ0  = ρ(a)η0 |η0 ,

∀ a ∈ A.

(A.2.4)

Proof Assume that π and ρ are unitarily equivalent and let U be the unitary operator, which establishes the equivalence. We shall prove that η0 = U ξ0 is strongly-cyclic for ρ. Let η ∈ Dρ . Then, there exists ξ ∈ Dπ and a net {aα } in A, such that U ξ = η and ξ = tπ -limα π(aα )ξ0 . This is equivalent to ∀ b ∈ A.

π(b)ξ = tπ - lim π(b)π(aα )ξ0 , α

Then, we have ρ(b)η = ρ(b)U ξ = U π(b)ξ = U tπ - lim π(b)π(aα )ξ0 α

= U tπ - lim π(baα )ξ0 = tπ - lim ρ(baα )U ξ0 α

α

= tπ - lim ρ(b)ρ(aα )U ξ0 = tπ - lim ρ(b)ρ(aα )η0 . α

α

This implies that η0 is strongly-cyclic for ρ. Moreover, ρ(a)η0 |η0  = ρ(a)U ξ0 |U ξ0  = U π(a)ξ0 |U ξ0  = π(a)ξ0 |ξ0 ,

∀ a ∈ A.

Conversely, assume that ρ is strongly-cyclic with strongly-cyclic vector η0 satisfying (A.2.4). We begin with defining U ξ0 := η0 and U π(a)ξ0 := ρ(a)η0 ,

a ∈ A.

The equality ρ(a)η0 2 = ρ(a)η0 |ρ(a)η0  = ρ(a ∗ a)η0 |η0  = π(a ∗ a)ξ |ξ  = π(a)ξ |π(a)ξ  0

= π(a)ξ0 

2

0

0

0

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proves at once that U is well-defined and isometric. Since {π(a)ξ0 : a ∈ A} is dense in Hπ , U extends to an isometric operator U from Hπ into Hρ . Since U is clearly invertible and the inverse is also isometric, U is unitary. By an argument similar to that used in the first part of the proof, one can easily show that U Dπ = Dρ . Finally, we have U π(a)π(b)ξ0 = U π(ab)ξ0 = ρ(ab)U ξ0 = ρ(a)ρ(b)η0 = ρ(a)U π(b)ξ0 ,

∀ a, b ∈ A.

Hence U π(a)ξ = ρ(a)U ξ , for every ξ ∈ {π(b)ξ0 : b ∈ A}. This easily implies that U π(a)ξ = ρ(a)U ξ , for every a ∈ A and ξ ∈ Dπ .  

A.2.1 Hermitian and Positive Linear Functionals Let now π be a *-representation of A with domain Dπ ⊆ Hπ . Fix an element ξ ∈ Dπ and define ωξ (a) := π(a)ξ |ξ ,

a ∈ A.

Then, ωξ is a linear functional on A with the properties: • ωξ (a ∗ ) = π(a ∗ )ξ |ξ  = ξ |π(a)ξ  = π(a)ξ |ξ  = ωξ (a), ∀ a ∈ A; • ωξ (a ∗ a) = π(a ∗ a)ξ |ξ  = π(a ∗ )π(a)ξ |ξ  = π(a)ξ |π(a)ξ  ≥ 0, ∀ a ∈ A; • |ωξ (a)|2 = |π(a)ξ |ξ |2 ≤ π(a)ξ 2 ξ 2 = ξ 2 ωξ (a ∗ a), ∀ a ∈ A. These properties of ωξ suggest the following Definition A.2.12 Let A be a *-algebra. A linear functional ω on A is called hermitian if ω(a ∗ ) = ω(a), for every a ∈ A; positive if ω(a ∗ a) ≥ 0, for all a ∈ A. A positive linear functional ω is called • faithful if a ∈ A and ω(a ∗ a) = 0 imply a = 0. • Hilbert bounded if there exists γ > 0, such that |ω(a)|2 ≤ γ ω(a ∗ a), for every

a ∈ A. • If A has a unit e, ω is called a state if ω(e) = 1.

Clearly, if ω is Hilbert bounded, then it is automatically positive. Moreover, in this case, one can put (see [19, Definition 9.4.2])   ωH = sup |ω(a)|2 : a ∈ A, ω(a ∗ a) = 1 . Every positive linear functional ω on A satisfies the equality, ω(b∗ a) = ω(a ∗ b),

∀ a, b ∈ A,

(A.2.5)

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as well as the Cauchy–Schwarz inequality |ω(b∗ a)|2 ≤ ω(a ∗ a)ω(b∗ b),

∀ a, b ∈ A.

(A.2.6)

To prove (A.2.5), let α ∈ C. Then,

0 ≤ ω (αa + b)∗ (αa + b) = |α|2 ω(a ∗ a) + αω(b∗ a) + αω(a ∗ b) + ω(b∗ b). Since |α|2 ω(a ∗ a) + ω(b∗ b) is real, αω(b∗ a) + αω(a ∗ b) must also be real, for every α ∈ C. This implies that ω(b∗ a) = ω(a ∗ b), for all a, b in A. As for the Cauchy–Schwarz inequality, for α ∈ R and a, b ∈ A, we have

0 ≤ ω (αa + b)∗ (αa + b) = α 2 ω(a ∗ a) + 2α (ω(b∗ a) + ω(b∗ b) ≤ α 2 ω(a ∗ a) + 2α|ω(b∗ a)| + ω(b∗ b), where ( means ‘real part’. For the last term to be nonnegative for any α ∈ R, the discriminant cannot be positive, i.e., |ω(b∗ a)|2 − ω(a ∗ a)ω(b∗ b) ≤ 0. As a direct consequence of (A.2.5) and (A.2.6), we have Proposition A.2.13 If A has a unit e and ω is a positive linear functional of A, then ω(a ∗ ) = ω(a) and |ω(a)|2 ≤ ω(e)ω(a ∗ a),

∀ a ∈ A;

i.e., ω is hermitian and Hilbert bounded with γ = ω(e). Remark A.2.14 If A has no unit, then ω may fail to be hermitian. If ω is hermitian, then clearly (A.2.5) holds. Remark A.2.15 If ω is a linear functional on A, then ω has a natural extension ωe to the unitization Ae of A (Remark A.1.1), obtained by setting:

ωe (a, λ) := ω(a) + λ,

∀ (a, λ) ∈ Ae .

It is easily seen that if ω is hermitian on A, then ωe is also hermitian on Ae ; but, if ω is positive on A, then ωe need not be positive on Ae . Nevertheless, an extension of a positive linear functional ω on A can be achieved under certain conditions and such conditions are given in Theorems A.2.19 and A.2.22. Since the functional ωξ defined, at the beginning of this section from a *representation π of A, is the prototype of a positive linear functional on A, it is natural to pose the question as to whether any positive linear functional can be

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realized as a vector functional ωξ ; more precisely, does there exist a *-representation πω and a vector ξω ∈ D(πω ), such that ω(a) = πω (a)ξω |ξω , for every a ∈ A? We shall discuss this question in the next subsections.

A.2.2 The Gelfand–Naimark–Segal Theorem Let ω be a positive linear functional on a *-algebra A. Let   Iω = a ∈ A : ω(a ∗ a) = 0 . Then, Iω is a left-ideal of A. Indeed, if a ∈ Iω and b ∈ A, the Cauchy–Schwarz inequality gives

ω(a ∗ b∗ ba)2 ≤ ω(a ∗ a)ω (b∗ ba)∗ b∗ ba = 0. Thus, ba ∈ Iω . We denote by Dω the quotient space A/Iω and with λω (a) the equivalent class of a ∈ A. Then, Dω is a pre-Hilbert space with inner product defined by λω (a)|λω (b) := ω(b∗ a),

a, b ∈ A.

Let Hω denote the Hilbert space completion of Dω . It is easily checked that a ∈ A → λω (a) ∈ Hω is a linear map. We now put πω◦ (a)λω (b) := λω (ab),

a, b ∈ A.

Then, for each a ∈ A, πω◦ (a) is well-defined. Indeed if λω (b) = 0, then clearly λω (ab) = 0, since Iω is a left-ideal. Moreover, we have



πω◦ (a)λω (b)|λω (c) = ω c∗ (ab) = ω (a ∗ c)∗ b = λω (b)|πω◦ (a ∗ )λω (c), with c ∈ A. This implies that πω◦ (a ∗ ) = πω◦ (a)† , for all a ∈ A. Therefore, πω◦ is a *-representation of A in Hω . Its closure πω is a closed *-representation of A, which is called the Gelfand–Naimark–Segal (for short, GNS) representation of A constructed by ω. Of course, πω satisfies the relation ω(c∗ ab) = πω (a)λω (b)|λω (c),

∀ a, b, c ∈ A.

  Suppose now that A has a unit e. Then, we have that πω◦ (a)λω (e) : a ∈ A = Dω ; hence, πω◦ is algebraically cyclic. The vector ξω := λω (e) is a strongly-cyclic   vector for πω ; i.e., πω (a)λω (e) : a ∈ A is dense in Dπω , with respect to the

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topology defined by the seminorms (A.2.1). In this case, one has ω(a) = π(a)ξω |ξω . Summing up, we have the following Theorem A.2.16 For every positive linear functional ω on a *-algebra A, there exists a Hilbert space Hω , a linear map λω : A → Hω and a closed *-representation πω of A with domain Dπω ⊆ Hω , such that ω(c∗ ab) = πω (a)λω (b)|λω (c),

∀ a, b, c ∈ A.

The triple (πω , λω , Hω ) is called the GNS construction for ω. If A has a unit e, then πω is strongly-cyclic with strongly-cyclic vector ξω = λω (e) and ω(a) = πω (a)ξω |ξω ,

∀ a ∈ A.

Moreover, if ρ is another closed strongly-cyclic *-representation of A with domain Dρ ⊆ Hρ and ηω a strongly-cyclic vector in Dρ , such that ω(a) = ρ(a)ηω |ηω , then ρ is unitarily equivalent to πω . The statement about the essential uniqueness of the GNS construction for *-algebras with unit follows from Theorem A.2.11. As seen above, there is some difference in the GNS theorem when considering algebras with unit or without unit. As we know, every *-algebra A without unit has a unitization Ae ; but a positive linear functional ω does not extend in a natural way to Ae , in general. We shall now examine, in more details, this problem. Definition A.2.17 Let A be a *-algebra. A positive linear functional ω on A is called extensible if ω is the restriction to A of some positive linear functional ω on Ae , the unitization of A. It is clear that if ω is extensible, then it is Hilbert bounded. We will show that the converse also holds. To begin with, let us give the following Lemma A.2.18 Let ω be a positive linear functional on A. Let (πω , λω , Hω ) be the GNS construction for ω. Then, ω is Hilbert bounded, if and only if, there exists a vector ζω ∈ Hω , such that ω(a) = λω (a)|ζω ,

∀ a ∈ A.

(A.2.7) 1/2

In this case, the vector ζω is uniquely determined and ζω  = ωH .   Proof Assume that ω is Hilbert bounded. Let us define on Dω ≡ λω (a) : a ∈ A a linear functional f , as follows f (λω (a)) := ω(a),

a ∈ A.

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The functional f is well-defined since if ω(a ∗ a) = 0, its Hilbert boundedness implies that ω(a) = 0. On the other hand,

1/2 1/2 |f λω (a) | = |ω(a)| ≤ ωH ω(a ∗ a)1/2 = ωH λω (a),

a ∈ A.

Thus, f is bounded on Dω . Therefore, it can be uniquely extended to a bounded linear functional on Hω . By the Riesz lemma, there exists a vector ζω ∈ Hω , such that f (λω (a)) = ω(a) = λω (a)|ζω ,

∀ a ∈ A.

Conversely, assume that there exists ζω , such that (A.2.7) holds. Then, |ω(a)| = |λω (a)|ζω | ≤ λω (a) ζω  = ζω  ω(a ∗ a)1/2 ,

∀ a ∈ A. 1/2

Hence, ω is Hilbert bounded. The uniqueness of ζω and the equality ζω  = ωH   also follow from the Riesz lemma. Theorem A.2.19 Let ω be a positive linear functional on A and let (πω , λω , Hω ) be the GNS construction for ω. The following statements are equivalent: (i) ω is extensible; (ii) ω is hermitian and Hilbert bounded; (iii) ω is Hilbert bounded and ωe : Ae → C defined by

ωe (a, λ) = ω(a) + λγ ,

∀ (a, λ) ∈ Ae ,

is a positive linear functional on Ae , for every γ ≥ ωH ; (iv) there exists a *-representation π of A with domain Dπ and a vector ξ ∈ Dπ , such that ω(a) = π(a)ξ |ξ ,

∀ a ∈ A;

(v) there exists a vector ζω ∈ Hω , such that ω(a) = λω (a)|ζω ,

∀ a ∈ A.

(A.2.8)

Proof (i) ⇒ (ii) Assume that ω is a positive linear functional that extends ω to Ae . Then, since Ae has a unit, ω is hermitian and |ω(a)|2 = |ω (a)|2 ≤ ω (e)ω (a ∗ a) = ω (e)ω(a ∗ a),

∀ a ∈ A.

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(ii) ⇒ (iii) Let ω be hermitian and Hilbert bounded. Let γ ≥ ωH . Then, for every a ∈ A and λ ∈ C, we have

ωe (a, λ)∗ (a, λ) = ω(a ∗ a) + λω(a) + λω(a ∗ ) + γ |λ|2 |ω(a)|2 2 |λ| ω(a ∗ a)



1 1 1 1 = ω(a ∗ a) 2 + ω(a ∗ a)− 2 λω(a) ω(a ∗ a) 2 + ω(a ∗ a)− 2 λω(a) 2  1 1   = ω(a ∗ a) 2 + ω(a ∗ a)− 2 λω(a) ≥ 0. ≥ ω(a ∗ a) + λω(a) + λω(a ∗ ) +

Consequently, ωe is positive. To prove that (iii) ⇒ (iv) it suffices to consider the GNS representation of Ae constructed from ωe and restrict it to A. (iv) ⇒ (v) If π is a *-representation as described in (iv), then ω is Hilbert bounded, therefore the result follows from Lemma A.2.18. (v) ⇒ (i) From Lemma A.2.18 it follows that ω is Hilbert bounded and that ωH = ζω 2 . We define

ωe (a, λ) = ω(a) + λζω 2 ,

∀ a ∈ A, λ ∈ C.

Then, with the same computation made in (ii) ⇒ (iii) one shows that ωe is positive on Ae . Hence, ω is extensible.   Let ω be a positive linear functional on A. Let b ∈ A and put ωb (a) := ω(b∗ ab),

a ∈ A.

Then, it is easily seen that ωb is a positive linear functional on A, for every b ∈ A. Proposition A.2.20 For every b ∈ A, ω is extensible and ω  ≤ ω(b∗ b). b

b H

Moreover, ωb (a) = πω (a)λω (b)|λω (b),

a ∈ A.

Proof By the Cauchy–Schwarz inequality we have |ωb (a)|2 = |ω(b∗ ab)|2 ≤ ω(b∗ b)ω(b∗ a ∗ ab) = ω(b∗ b)ωb (a ∗ a),

∀ a ∈ A.

This shows at once that ωb is Hilbert bounded and that ωb H ≤ ω(b∗ b). The   statement then follows from Theorem A.2.19. Definition A.2.21 Let A be a *-algebra and ω a positive linear functional on A. We say that ω is representable if there exists a closed strongly-cyclic *-representation

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π , with strongly-cyclic vector ξ0 , such that ω(a) = π(a)ξ0 |ξ0 ,

∀ a ∈ A.

From the GNS theorem it follows easily that every positive linear functional on a *-algebra A with unit e is representable. In the general case we have the following. Theorem A.2.22 Let A be a *-algebra and ω a positive linear functional on A. The following statements are equivalent: (i) ω is representable; (ii) ω is Hilbert bounded; (iii) ω is extensible.  

Proof It is an easy consequence of Theorems A.2.19 and A.2.16.

A.2.3 Boundedness of the GNS Representation; Admissibility Proposition A.2.23 Let A be a *-algebra with unit e. Let ω be a positive linear functional on A. The following statements are equivalent: (1) the GNS representation πω constructed from ω is bounded; (2) for every a ∈ A, there exists a constant Ca , such that ω(b∗ a ∗ ab) ≤ Ca2 ω(b∗ b),

∀ b ∈ A.

(A.2.9)

Proof The boundedness of the GNS representation is equivalent to the following condition: For every a ∈ A, there exists a constant Ca , such that πω (a)λω (b)2 ≤ Ca2 λω (b)2 ,

∀ b ∈ A.

The statement then follows from the equalities πω (a)λω (b)2 = ω(b∗ a ∗ ab) and λω (b)2 = ω(b∗ b).

 

Definition A.2.24 Let A be a *-algebra with unit e and ω a positive linear functional on A. We say that ω is admissible if condition (2) of Proposition A.2.23 holds. Remark A.2.25 If ω is admissible, the smallest constant Ca satisfying (A.2.9) is easily determined. Indeed, since, for every a ∈ A, πω (a) is bounded, its closure πω (a) is a bounded linear operator on Hω . Hence, ω(b∗ a ∗ ab) = πω (a)λω (b)2 ≤ πω (a)2 λω (b)2 = πω (a)2 ω(b∗ b). Hence, πω (a) is the best constant for (A.2.9).

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Lemma A.2.26 (Kaplansky’s Inequality) Let ω be a positive linear functional on A. Then, for every n ∈ N, ω(b∗ a ∗ ab) ≤ ω(b∗ b)1−2

−n

 2−n n ω(b∗ (a ∗ a)2 b) ,

∀ a, b ∈ A.

Proof Using the Cauchy–Schwarz inequality, we get

1/2 ω(b∗ a ∗ ab) ≤ ω(b∗ b)1/2 ω b∗ (a ∗ a)2 b . Thus the statement is true for n = 1. Assume that it holds for n ∈ N. Since n n+1 1/2 , ω b∗ (a ∗ a)2 b ≤ ω(b∗ b)1/2 ω b∗ (a ∗ a)2 b we have ω(b∗ a ∗ ab) ≤ ω(b∗ b)1−2

−n

  −n n 2 ω b∗ (a ∗ a)2 b

≤ ω(b∗ b)1−2

−n

 2−n n+1 ω(b∗ b)1/2 ω(b∗ (a ∗ a)2 b)1/2

= ω(b∗ b)1−2

−(n+1)

  −(n+1) n+1 2 ω b∗ (a ∗ a)2 b .

 

Definition A.2.27 Let p be a seminorm on A. We say that ω is relatively p-bounded if ∀ b ∈ A, ∃ γb > 0, such that |ωb (a)| ≤ γb p(a),

∀ a ∈ A.

We say that ω is p-bounded if ∃ γ > 0, such that |ω(a)| ≤ γ p(a),

∀ a ∈ A.

Remark A.2.28 If p is submultiplicative (i.e., p(ab) ≤ p(a)p(b), for every a, b ∈ A), then p-boundedness implies relative p-boundedness. If A has a unit e, relative pboundedness implies p-boundedness. So, if A has a unit and p is submultiplicative, relative p-boundedness and p-boundedness are equivalent. Proposition A.2.29 Let p be a seminorm on A, such that n 2−n Ca2 := lim inf p (a ∗ a)2 < ∞, n→∞

Then, if ω is relatively p-bounded, ω is admissible.

∀ a ∈ A.

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Proof By Kaplansky’s inequality, for every a, b ∈ A, ωb (a ∗ a) ≤ ω(b∗ b)1−2

−n

≤ ω(b∗ b)1−2

−n

  −n n 2 ωb (a ∗ a)2 γb2

−n

n 2−n p (a ∗ a)2 .

Taking the lim inf of the right hand side, we obtain ω(b∗ a ∗ ab) ≤ Ca2 ω(b∗ b)  

and this completes the proof. Corollary A.2.30 If p is submultiplicative, is relatively p-bounded, then

p(a ∗ )

ω(b∗ a ∗ ab) ≤ p(a)2 ω(b∗ b),

≤ p(a), for every a in A and ω ∀ a, b ∈ A.

Proof In this case, we have, in fact n 2−n lim inf p (a ∗ a)2 ≤ lim inf p(a ∗ a) ≤ p(a)2 , n→∞

n→∞

∀ a ∈ A.

 

Definition A.2.31 Let A be a *-algebra. A seminorm p on A is called a C*seminorm if (1) p(ab) ≤ p(a)p(b), for all a, b ∈ A; (2) p(a ∗ a) = p(a)2 , for each a ∈ A. Remark A.2.32 We remark that by a beautiful result of Sebestyén, (2) always implies (1). We also notice that (1) and (2) imply p(a ∗ ) = p(a), for each a ∈ A. Indeed, one has that p(a)2 = p(a ∗ a) ≤ p(a ∗ )p(a), therefore p(a) ≤ p(a ∗ ); the result then follows by interchanging the roles of a and a ∗ . Proposition A.2.33 Suppose that A admits a bounded *-representation π . Then, p(a) := π(a), for every a ∈ A, is a C*-seminorm on A. The proof is straightforward. Proposition A.2.34 Let ω be a positive linear functional on a *-algebra A with unit e. Then, ω is admissible, if and only if, there exists a C*-seminorm p on A, such that ω is relatively p-bounded. Proof If there exists a C*-seminorm p on A, such that ω is relatively p-bounded, then by Corollary A.2.30 we get the assertion. Conversely, assume that ω is admissible. Then, by Remark A.2.25, ω(b∗ a ∗ ab) ≤ πω (a)2 ω(b∗ b),

∀ a ∈ A.

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Thus, taking also into account Proposition A.2.13 , we obtain |ωb (a)| ≤ ωb (e)1/2 ωb (a ∗ a)1/2 ≤ ωb (e)1/2 ω(b∗ b)1/2 πω (a) = γb p(a),

∀ a ∈ A,

where γb ≡ ωb (e)1/2 ω(b∗ b)1/2 and p(a) := πω (a), a ∈ A. The assertion now follows from Definition A.2.27 and the fact that p(·) is a C*-seminorm.   It is quite clear that the notions of admissibility and representability for a positive linear functional are independent. Example A.2.35 Consider the *-algebra, say A, of complex polynomials with pointwise algebraic operations and involution by the complex conjugation. Define 

∞

ω(p) =

p(x)e−x dx, p ∈ A.

0

Then, ω is positive and representable, since A has a unit. On the other hand, if we take p(x) = x and q(x) = x n , we get ω(p2 q 2 ) (2n + 2)! = = (2n + 2)(2n + 1). 2 (2n)! ω(q ) Hence, ω is not admissible. In Sect. A.4, we will study in more details the admissibility of positive linear functionals for normed *-algebras. Before doing this, we need, however, a deeper knowledge of the structure of normed or Banach *-algebras.

A.3 Spectral Radius, Spectrum and All That Definition A.3.1 Let A be a normed algebra. For a in A define ν(a) := lim sup a n 1/n . n→∞

Proposition A.3.2 Let a, b be elements in a normed algebra A, and α ∈ C. Then,   (1) ν(a) = inf a n 1/n : n ∈ N ; (2) 0 ≤ ν(a) ≤ a; (3) ν(αa) = |α|ν(a); (4) ν(ab) = ν(ba) and ν(a k ) = ν(a)k , k ∈ N; (5) If ab = ba, then ν(a + b) ≤ ν(a) + ν(b) and ν(ab) ≤ ν(a)ν(b). Proof We shallprove only (1).  Put ν = inf a n 1/n : n ∈ N ; we shall prove that ν = limn→∞ a n 1/n . Let > 0; then, there exists m ∈ N, such that a m 1/m < ν + . For n > m we can

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write n = pm + q with 0 ≤ q ≤ m − 1. Since q/m → 0, pm/n → 1. Then, we have a n 1/n = a pm+q 1/n ≤ a m p/n aq/n < (ν + )pm/n aq/n → ν + . This implies that lim supn→∞ a n 1/n < ν + ; from the arbitrariness of , we get lim sup a n 1/n ≤ ν. n→∞

On the other hand, ν ≤ a n 1/n ; therefore, ν ≤ lim infn→∞ a n 1/n . In conclusion, lim a n 1/n = ν.

 

n→∞

Proposition A.3.3 Let a be an element of a normed algebra A, such that a2 . Then, ν(a) = a. k

a 2 

=

k

Proof If a 2  = a2 , then a 2  = a2 , k ∈ N. Therefore, ν(a) = k k   limk→∞ a 2 1/2 = a. Proposition A.3.4 The equality ν(a) = a holds, for every a ∈ A, if and only if, a 2  = a2 , for every a ∈ A. Proof The sufficiency follows from Proposition A.3.3. As for the necessity, if ν(a) = a, for every a ∈ A, we have a 2  = ν(a 2 ) = ν(a)2 = a2 .

 

In a C*-algebra every hermitian element a, i.e., a = a ∗ , satisfies the condition a 2  = a2 . Thus, one has the following Corollary A.3.5 Let A be a C*-algebra. For every a ∈ A, such that a = a ∗ , ν(a) = a.

Remark A.3.6 If ab = ba, then ν(a + b) ≤ ν(a) + ν(b) and ν(ab) ≤ ν(a)ν(b). Therefore, if A is commutative, the function ν(·) is a seminorm. If, in addition, a 2  = a2 , for every a ∈ A, then ν(·) is a norm. It can be proved that a normed algebra A, where a 2  = a2 , for every a ∈ A, is necessarily commutative (see, for instance, [8, p. 345, (B.6.16) Theorem]). Let A be a *-algebra with unit e. We say that an element a ∈ A has a right inverse if there exists b ∈ A, such that ab = e. Similarly, a has a left inverse if there exists c ∈ A, such that ca = e. An element a ∈ A is called invertible if it has both left and right inverse. In this case, the left inverse c and the right inverse b of a coincide. Indeed, c = ce = c(ab) = (ca)b = eb = b. The inverse of a is denoted, as usual, by a −1 . The set G of all invertible elements of A constitutes a multiplicative group. Proposition A.3.7 Let A be a Banach algebra with unit e. If a ∈ A with ν(e − a) < n 1, then a is invertible and a −1 = e + ∞ n=1 (e − a) .

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n 1/n < 1. Thus, the numerical series Proof ∞ If ν(e −na) < 1, then limn→∞ (e − a)  ∞ n n=0 (e−a)  is convergent. This implies that the series n=0 (e−a) converges −1 absolutely to an element b ∈ A. We prove that a = e + b. Indeed,

a(e+b) = a+

∞  n=1

a(e−a)n = a+

∞ 

(a−e)(e−a)n +

n=1

∞ 

(e−a)n = a+(e−a) = e.

n=1

In a similar way one also proves that (e + b)a = e.

 

Since, for b ∈ A, one has ν(b) ≤ b, we get Corollary A.3.8 If a ∈ A, where A is a Banach algebra with unit, and e−a < 1, then a is invertible. With an obvious substitution in Proposition A.3.7, one also has the following Corollary A.3.9 Let A be a Banach algebra with unit e. If a ∈ A with ν(a) < 1, n then e − a is invertible and (e − a)−1 = e + ∞ n=1 a . Theorem A.3.10 If A is a Banach algebra with unit e, then the group G of all invertible elements of A is open in A. Moreover, the map a → a −1 is continuous on G.   1 Proof Let a ∈ G. The set Ua = b ∈ A : a − b < a −1 is obviously a  neighborhood of a. We prove that Ua ⊂ G. Indeed, if b ∈ Ua , one has e − a −1 b = a −1 (a − b) ≤ a −1  a − b < 1. From Corollary A.3.8, a −1 b is invertible and this implies that b is also invertible. As for the continuity of the inversion, we can use the identity b−1 − a −1 = a −1 (a − b)a −1 + (b−1 − a −1 )(a − b)a −1 which implies b−1 − a −1  ≤ a −1 2 b − a + b−1 − a −1 b − aa −1 . Thus b−1 − a −1  ≤

a −1 2 b − a 1 − b − aa −1 

and the right hand side converges to 0 if b → a.

 

One of the most important and useful concepts in the theory of Banach algebras is the notion of the spectrum of an element.

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Definition A.3.11 Let a be an element of an algebra A with unit e. The resolvent set of a is the following subset of the complex plane C   ρA (a) = λ ∈ C : a − λe is invertible in A . The spectrum of a is the set σA (a) = C \ ρA (a). In general, σA (·) is neither non-empty, nor bounded (see, e.g., [9, 4.3 Examples]). Theorems A.3.14 and A.3.15 give that context, where both properties always hold. Remark A.3.12 Clearly if B is a subalgebra of A, we may have ρB (a) = ρA (a) (and consequently σB (a) = σA (a)). For this reason, we have explicitly mentioned the dependence on A in the notation. The following statement can be easily proved. Proposition A.3.13 If B ⊂ A is a subalgebra containing the unit of A then, for each a ∈ B, σA (a) ⊆ σB (a). Theorem A.3.14 If a is an element of a normed algebra A with unit e, then σA (a) is nonempty. Proof When it is not so, the inverse (a − λe)−1 would exist, for every λ ∈ C. In particular, the element a −1 exists. By the Hahn–Banach theorem there is a bounded linear functional on A, such that (a −1 ) = 1. Let a(λ) = (a − λe)−1 and f (λ) = (a(λ)). Since (a −1 ) = 1, f (0) = 1. We shall prove that f is analytic on the whole plane C. Indeed, the identity a(λ) − a(μ) = (λ − μ)a(λ)a(μ), λ, μ ∈ C, implies that f (λ) − f (μ) = lim (a(λ)a(μ)) = (a(μ)2 ). λ→μ λ→μ λ−μ lim

Since, |f (λ)| ≤  a(λ),

λ∈C



−1 and a(λ) = λ−1 λ−1 a − e , with a(λ) → 0, as |λ| → ∞ (take into account the continuity of the inverse map), it follows that |f (λ)| is bounded. On the other hand, by Liouville’s theorem the function f is constant; hence f = 0, which is impossible since f (0) = 1.   Theorem A.3.15 Let A be a Banach algebra with unit and a ∈ A. Then, (1) |λ| ≤ a, for every λ ∈ σA (a); (2) σA (a) is a nonempty compact subset of C.

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Proof 1. Let λ ∈ σA (a) with |λ| > a. Then, λ−1 a < 1, hence by Corollary A.3.8 the element e − λ−1 a is invertible and clearly the same is also true for the element λe − a. But, this means that λ ∈ σA (a), a contradiction. 2. We already know that σA (a) is nonempty and bounded. We now prove that it is closed. The map f : λ ∈ C → a − λe ∈ A is continuous. Since the set G of invertible elements is open, the set f −1 (G ∩ Im f ) is also open, therefore its complement is closed. But, this is exactly σA (a).   Theorem A.3.16 (Mazur–Gelfand) Let A be a normed algebra with unit e. Assume that each nonzero element of A is invertible (i.e., A is a division algebra). Then, A is isometrically isomorphic to C. Proof It suffices to show that the map λ ∈ C → λe ∈ A is surjective. Let a ∈ A be nonzero. By Theorem A.3.14, there exists λ ∈ C, such that a − λe is not invertible. Since each nonzero element in A has an inverse, it must be a − λe = 0, i.e.,   a = λe. Remark A.3.17 We warn the reader that not too much regularity can be imposed to a Banach algebra without trivializing it. The Mazur–Gelfand theorem already implies the existence of non invertible elements in a non trivial normed algebra. Similarly, in a Banach algebra A with unit, the following two facts hold (see, for instance, [8, (B.4.7) Proposition]) 1. If a −1  = a−1 , for every a ∈ G, then A is topologically isomorphic to C. 2. If ab = ab, for all a, b ∈ A, then A is topologically isomorphic to C.  n Lemma A.3.18 Let A be a Banach algebra with unit e and ∞ n=0 βn z be a power series with radius of convergence R. Let f  (z) be the sum of the series. Then, for n each a ∈ A, with a ≤ r < R, the series ∞ n=0 βn a converges to an element of A that we call f (a). Moreover, if λ ∈ σA (a) and |λ| < r, then f (λ) ∈ σA (f (a)). Proof For M > N , we have  M  M M      n βn a  ≤ |βn |a n  ≤ |βn |r n → 0    n=N

n=N

n=N

as N, M → ∞. To prove the second statement we proceed as follows: f (a) − f (λ)e =

∞ 

βn (a n − λn e)

n=0

=

∞ 

βn (a − λe)

n=0

= (a − λe)

n−1 

λk a n−k−1

k=0 ∞  n=0

βn

n−1  k=0

λk a n−k−1 .

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   k a n−k−1  ≤ nr n−1 , the latter series converges to some b ∈ A. Thus, Since,  n−1 λ  k=0 f (a) − f (λ)e = (a − λe)b. If f (a) − f (λ)e has an inverse, then a − λe also has an inverse. This completes the proof.   In the next Lemma, we list some easy relations, of purely algebraic nature, that hold for the spectrum. Lemma A.3.19 Let A be a *-algebra with unit e. For a, b ∈ A and β ∈ C, we have   (1) σA (a − βe) = λ − β : λ ∈ σA (a) ;   (2) σA (a ∗ ) = λ : λ ∈ σA (a) ; (3) σA (ab) ∪ {0} = σA (ba) ∪ {0};  (4) If a −1 exists, then σA (a −1 ) = λ−1 : λ ∈ σA (a) . Proof For proving (3), we take into account that if λ ∈ σA (ba), then



(ab − λe) a(ba − λe)−1 b − e = λe = a(ba − λe)−1 b − e (ab − λe). This implies that (ab − λe) is invertible, at least if λ = 0. Thus, σA (ab) ∪ {0} ⊆ σA (ba) ∪ {0}. Interchanging the roles of a and b one obtains the opposite   inclusion. The statement (3) has the following very important consequence: Theorem A.3.20 (Wielandt–Wintner) Let A be a Banach algebra with unit e. Then, there are no elements a, b ∈ A, such that ab − ba = e. Proof Assume that two such elements exist. From Theorem A.3.15 σA (ab) and σA (ba) are non empty subsets of C. By (3) of Lemma A.3.19, σA (ab) and σA (ba) differ at most by 0. But, this is impossible, since the assumption and (1) of Lemma A.3.19 imply that σA (ab) = 1 + σA (ba).   Theorem A.3.20 shows the existence of algebras that do not have bounded representations. Indeed, let us consider on Cc∞ (R) the operators a = d/dx and b : f → idR f , x ∈ R, where idR is the identity map of R. Then, it is easily seen that (ab − ba)f (x) = f (x); i.e., ab − ba = e. Thus, the algebra A of operators on Cc∞ (R) generated by a, b has no bounded representations. Definition A.3.21 Let A be an algebra. The spectral radius of an element a ∈ A is defined as   |a|σ = sup |λ| : λ ∈ σA (a) . In general, one has that 0 ≤ |a|σ ≤ +∞. The next theorem, due to Beurling and Gelfand, shows that in a Banach algebra the spectral radius of an element is always finite; for ν(·), see Definition A.3.1.

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Theorem A.3.22 Let A be a Banach algebra with unit e and a ∈ A. Then, |a|σ = lim a n 1/n = ν(a). n→∞

Proof If a ∈ A and λ ∈ σA (a), then by Theorem A.3.15, |λ| ≤ a and by Lemma A.3.18, λn ∈ σA (a n ). This implies |a|nσ ≤ |a n |σ . Then, |a|σ ≤ (|a n |σ )1/n ≤ a n 1/n , and so |a|σ ≤ lim a n 1/n = ν(a). n→∞

To prove the converse, we make use of an argument of complex variables. If r > |a|σ , then the function f (λ) = (λe − a)−1 is analytic in the exterior of the circle γr of radius r. In this region, it has the following Laurent’s expansion f (λ) =

 ∞  an 1 n , with a = λn f (λ)dλ. 2π i γr λn+1 n=0

Now, put M(r) = sup0≤θ≤2π f (reiθ . Then, a n  ≤ r n+1 M(r) = r n (rM(r)) and so

lim a n 1/n ≤ r.

n→∞

Since this is true for any r > |a|σ , we finally get lim a n 1/n ≤ |a|σ .

n→∞

 

Definition A.3.23 Let A be a Banach *-algebra with unit e. An element a ∈ A is called positive if a ∗ = a and σA (a) ⊂ R+ ∪ {0}. Positive elements, as well as positive linear functionals, play a crucial role in the theory of Banach *-algebras. Proposition A.3.24 Let A be a C*-algebra with unit e and let a = a ∗ ∈ A. The following statements are equivalent: (1) (2) (3) (4)

σA (a) ⊂ R+ ∪ {0} (i.e., a is positive); a = b∗ b, for some b ∈ A; a = h2 , for some h ∈ A with h∗ = h; if a ≤ α, then αe − a ≤ α.

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Remark A.3.25 Generally, in a Banach *-algebra there is no equivalence between the four conditions above. We only have (1) ⇒ (2), (1) ⇒ (4).

A.4 Admissible Positive Linear Functionals Lemma A.4.1 Let A be a Banach *-algebra with unit e. For any a ∈ A with a = a ∗ and ν(a) < 1, there exists an element b ∈ A with b = b∗ , such that 2b − b2 = a. √ Proof The basic idea of the proof is to try to define the element b = e + e − a, which formally solves the given equation. As it is known from the elementary calculus, if |λ| < 1, the series −

! ∞  1/2 (−λ)n , n k=1

converges to a solution ζ of the equation 2z − z2 = λ. We can apply Lemma A.3.18, taking into account that ν(a) = a, since a = a ∗ . Thus, the series −

! ∞  1/2 (−a)n , n k=1

converges to an element b ∈ A, which satisfies the desired equation. The fact that b = b∗ follows from the continuity of the involution.   Theorem A.4.2 Let A be a Banach *-algebra with unit e and ω a positive linear functional on A. Then, |ω(b∗ hb)| ≤ ν(h)ω(b∗ b),

∀ b ∈ A, h = h∗ ∈ A.

Proof Assume that ν(h) < 1. By Lemma A.4.1, there exist elements r, s ∈ A with r = r ∗ , s = s ∗ , such that 2r − r 2 = h , 2s − s 2 = −h. For b ∈ A we put x = (e − r)b, y = (e − s)b. Then, it is easily seen that x ∗ x = b∗ (e − r)2 b = b∗ (e − h)b y ∗ y = b∗ (e − s)2 b = b∗ (e + h)b. Then, ω(b∗ (e − h)b) ≥ 0 and ω(b∗ (e + h)b) ≥ 0. These inequalities imply that |ω(b∗ hb)| ≤ ω(b∗ b),

(A.4.1)

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in the case ν(h) < 1. For an arbitrary h = h∗ ∈ A and for > 0, we put k = h/(ν(h) + ); thus ν(k) < 1. Applying the inequality (A.4.1), we obtain |ω(b∗ hb)| ≤ (ν(h) + )ω(b∗ b). Since is arbitrary, we get the result.

 

From Definition A.2.24 and the previous theorem it follows immediately that Corollary A.4.3 Every positive linear functional on a Banach *-algebra with unit is admissible. But there is something more! Theorem A.4.4 Every positive linear functional ω on a Banach *-algebra A with unit e is bounded and ω = ω(e). Proof Indeed, applying Theorem A.4.2 with b = e and h = a ∗ a, a ∈ A, we obtain ω(a ∗ a) ≤ ω(e)ν(a ∗ a). Then, a simple application of the Cauchy–Schwarz inequality gives |ω(a)| ≤ ω(e)ν(a ∗ a)1/2 , a ∈ A. Taking into account that ν(a ∗ a) = a ∗ a, we obtain |ω(a)| ≤ ω(e)a ∗ a1/2 ≤ ω(e)a, a ∈ A; thus ω is bounded and ω ≤ ω(e), but evidently ω(e) ≤ ω too, therefore the equality holds.   The following theorem gives the converse of the previous one for C*-algebras. Theorem A.4.5 Let A be a C*-algebra with unit e. Every continuous linear functional ω on A, such that ω = ω(e) is positive. Proof First of all, we prove that ω(a ∗ a) is real, for every a ∈ A. If ω : A → C is a linear functional, define ω∗ : A → C, such that ω∗ (a) := ω(a ∗ ), for all a ∈ A. Then, ω∗ is a linear functional on A, called adjoint of ω. If ω∗ = ω, that is ω(a ∗ ) = ω(a), for all a ∈ A, then ω is called hermitian (see also Sect. A.2.1). Each linear functional ω can be written as ω = ω1 + iω2 , with ω1 , ω2 hermitian.

It suffices to put ω1 (a) = 12 ω(a) + ω(a ∗ ) and ω2 (a) = 2i1 ω(a) − ω(a ∗ ) , a ∈ A. We assume that ω(e) = 1. Then, ω1 (e) = 1 and ω2 (e) = 0. Moreover, ω1 (a ∗ a) is real, for every a ∈ A. We shall show that ω2 is on the whole A zero. Let h = h∗ ∈ A; define x = αe − ih, where α is a real number. Then, x2 = α 2 e + h2  ≤ α 2 + h2 .

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The equality |ω(x)|2 = α 2 + 2αω2 (h) + ω1 (h)2 + ω2 (h)2 implies that α 2 ≤ |ω(x)|2 − 2αω2 (h). Hence, x2 ≤ |ω(x)|2 − 2αω2 (h) + h2 ≤ x2 − 2αω2 (h) + h2 . In conclusion, 2αω2 (h) ≤ h2 . This inequality holds for any α ∈ R. Thus, ω2 (h) = 0, for all h = h∗ ∈ A. This easily implies that ω2 is identically zero. Hence, ω(a ∗ a) is real, for every a ∈ A. To prove the positivity of ω, we assume that there exists a ∈ A, such that a ∗ a < 1 and ω(a ∗ a) < 0. By Lemma A.4.1, there exists b = b∗ ∈ A, such that a ∗ a = 2b − b2 ; thus e − a ∗ a = (e − b)2 , i.e., ω(e − a ∗ a) is positive. Therefore, by our assumption and Proposition A.3.24, we have 1 = ω(e) = ω(e − a ∗ a) + ω(a ∗ a) < ω(e − a ∗ a) ≤ e − a ∗ a ≤ 1.  

But, this is a contradiction.

Remark A.4.6 One may wonder if any hermitian linear functional on a C*-algebra is also continuous. The answer is negative as the following example shows. Let A be a C*-algebra and {xn } a sequence of linearly independent elements of  the real vector subspace Ah := x ∈ A : x = x ∗ . Assume xn  = 1, for every natural number n. Let E = yα : α ∈  be a Hamel basis for Ah containing the sequence {xn }. Clearly E is a Hamel basis for A, too. Now define  f (x) =

n, x = xn 0, x ∈ E \ {xn }.

Extend f by linearity to the whole algebra  A. Then, fisn a linear functional on A and it is hermitian. Indeed, let x = α∈F λα yα + k=1 μk xk , where F is a finite subset of  and y =  x , for every α ∈ F and k ∈ {1, . . . , n}. Then, x ∗ = α k  n α∈F λα yα + k=1 μk xk . Hence, f (x) =

n  k=1

μk · k and f (x ∗ ) =

n 

μk · k.

k=1

Thus, f (x ∗ ) = f (x), ∀ x ∈ A; i.e., f is hermitian. The functional f is clearly not bounded, since xn  = 1, but f (xn ) = n → ∞.

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A.5 C*-Seminorms on Banach *-Algebras If A is a Banach *-algebra with unit e, Theorem A.4.4 allows us to construct a seminorm on A by putting p(a) = sup ω(a ∗ a)1/2 , ω∈S (A)

a ∈ A,

where S(A) denotes the set of all states on A; that is, of all positive linear functionals ω on A with ω(e) = 1. Then, p(a) ≤ a, for every a ∈ A. We notice that if ω is a positive linear functional on A and b ∈ A, then the linear functional ωb defined before Proposition A.2.20 is also positive. Moreover, if ω(b∗ b) = 1, then ωb ∈ S(A). Proposition A.5.1 If A is a Banach *-algebra with unit e, then p(a) := sup ω(a ∗ a)1/2 , ω∈S (A)

defines a C*-seminorm on A. Proof We prove (1) of Definition A.2.31. We follow [95]. Let a, b ∈ A and c = ab. Then, if ω ∈ S(A), ω(c∗ c) = ωb (a ∗ a). If ω(b∗ b) = 0, then by the Cauchy– Schwarz inequality, ω(c∗ c) = 0, so that inequality (1) trivially holds. Now suppose that ω(b∗ b) > 0 and put x = b/ω(b∗ b)1/2 . It follows that ωx ∈ S(A) and ω(c∗ c) = ωx (a ∗ a)ω(b∗ b), therefore ω(c∗ c) ≤ p(a)2 p(b)2 . Taking sup on the left hand side, we get the result. As for (2) of Definition A.2.31, if a ∈ A and ω ∈ S(A), we have ω(a ∗ a)2 ≤ ω(a ∗ aa ∗ a) ≤ p(a ∗ a)2 , where the first inequality is due to the Cauchy–Schwarz inequality (see, in particular, Proposition A.2.13) and the second to the definition of p. This implies that p(a)2 ≤ p(a ∗ a). The statement then follows from submultiplicativity of p that we have already proven.   Note that, in general, p is not a norm; i.e., it may happen that p(a) = 0, with 0 = a ∈ A. Let   Np = b ∈ A : p(b) = 0 . It is easily seen that Np is a closed *-ideal of A. Then, the quotient A/Np is a C*-algebra, under the norm [a] = p(a), where a ∈ A and [a] ≡ a + Np .

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Definition A.5.2 The *-radical R(A) of a Banach *-algebra A is the intersection of the kernels of all *-representations of A on some Hilbert space. The Banach *algebra A is called *-semisimple if its *-radical R(A) is {0} (in this regard, see also Definitions 3.1.21 and 3.1.23). Proposition A.5.3 The *-radical R(A) of a Banach *-algebra A with unit e coincides with Np . Proof If a ∈ R(A) and π is a *-representation of A on a Hilbert space, then π(a) = 0. This is, in particular true, for the GNS representation πω constructed by starting from an element ω ∈ S(A). Thus, πω (a)λω (e) = 0 and this implies that p(a) = 0. Conversely, we prove that if π(a) = 0, for some *-representation π , then a ∈ Np . Indeed, if π(a) = 0, there exists a vector ξ0 ∈ H, ξ0  = 1, such that π(a)ξ0 = 0. We define ω(a) := π(a)ξ0 |ξ0 ,

a ∈ A.

Since ω(a ∗ a) = π(a)ξ0 2 = 0 and ω is positive, it follows that a ∈ Np .

 

In what we have done so far, there is a question that remains up to now unsolved. Can we be sure that the set of positive linear functionals on A does not reduce only to {0}? We shall answer this question in the case where A is a C*-algebra. Proposition A.5.4 Let A be a C*-algebra with unit e. For each a ∈ A, there exists a positive linear functional ω on A, with ω(e) = 1 and ω(a ∗ a) = a2 .   Proof Let a ∈ A and B := αe + βa ∗ a : α, β ∈ C . Then, B is a vector subspace of A. We define a linear functional f on B by f (αe + βa ∗ a) := α + βa2 . Since |a ∗ a|σ = a ∗ a, a ∈ A, we have that

  |α + βa2 | ≤ sup |α + βλ| : λ ∈ σA (a ∗ a) ≤ αe + βa ∗ a. Therefore, f is continuous and f  ≤ 1; but f (e) = 1, hence f  = 1. By the Hahn–Banach theorem, f has an extension ω to the whole A with ω = 1. By Proposition A.4.5 it follows that ω is positive.   The following theorem, which is one of the crucial points of the theory of C*algebras, is now a simple consequence of A.5.4 (recall also that p(a) =: [a], a ∈ A). Theorem A.5.5 Let A be a C*-algebra with unit. Then, a = p(a) = sup ω(a ∗ a)1/2 , ω∈S (A)

∀ a ∈ A.

Remark A.5.6 The previous theorem, which also shows that any C*-algebra is *-semisimple, was first proved by Gelfand and Naimark; for this reason p is often called in the literature Gelfand–Naimark seminorm.

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A.6 Gelfand Theory We recall some basic definitions on ideals. Let A be an algebra. A vector subspace I of A is called a left ideal of A if ax ∈ I, for any a ∈ A and x ∈ I. A right ideal is similarly defined. A two-sided ideal, or simply ideal is a left ideal that is also a right ideal. If A is unital, a left ideal I in A is called maximal if I is not contained in any other proper left ideal of A. With the help of Zorn’s lemma it can be proved that each left ideal of A is contained in a maximal one. If A has no unit, then a left ideal I in A is called modular if there exists u ∈ A, such that a − au ∈ I, for all a ∈ A. In this case, u is called a right unit modulo I. The (Jacobson) radical of A, denoted by rad(A) is the intersection of all left maximal modular ideals of A. For more details, see [8, pp. 124, 125]). If rad(A) = {0}, then A is called semisimple.

A.6.1 The Commutative Case Let A be a commutative Banach algebra. A multiplicative linear functional or character of A is a non-zero linear functional ω on A that preserves multiplication, i.e., ω(ab) = ω(a)ω(b),

∀ a, b ∈ A.

Since ω is non-zero, it is surjective. Indeed, if 0 = α ∈ C and a ∈ A, with ω(a) = 0, then the element b = αa/ω(a) of A gives ω(b) = α. Summing up, each character ω of A is a non-trivial homomorphism of A onto C.  For the sake of simplicity, the set of all characters of A will be denoted by  A,

instead of M(A), as we did in Sect. 7.3.

Furthermore, note that if A has a unit e, then a C-valued homomorphism ω on A is a character, if and only if, ω(e) = 1. Proposition A.6.1 Let A be a commutative Banach algebra and ω ∈  A. Then, K = Ker ω is a maximal modular ideal of A. In particular, A is generated by K and an element b ∈ / K. Proof As we have seen, ω is surjective. Thus, A/K is isomorphic to C; hence A/K has a unit, i.e., there exists u ∈ A, such that (a + K)(u + K) = a + K, for all a ∈ A. This implies that a − au ∈ K. Therefore, K is modular. Now we prove that A is generated by K and by an element b ∈ K. If b ∈ K, then ω(b) = 0. For ω(a) a ∈ A, put c ≡ a − ω(a) ω(b) b. Then, ω(c) = 0 and clearly a = ω(b) b + c is the desired decomposition.  

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Theorem A.6.2 Let A be a commutative Banach algebra with unit e and ω ∈  A. Then, |ω(a)| ≤ a, for all a ∈ A. Therefore, ω is continuous and ω = 1. Proof Assume that there is a ∈ A, such that |ω(a)| > a; put λ = ω(a). Then, a/λ < 1, and so (e − a/λ) has an inverse b in A. Then, we have 1 = ω(e) = ω(b(e − a/λ)) = ω(b)ω(e − a/λ) = 0, a contradiction.

 

Remark A.6.3 By Proposition A.2.13, each character on a C*-algebra with unit is a positive linear functional. Proposition A.6.4 Let A be a commutative Banach algebra. Then, every maximal modular ideal of A is the kernel of a unique character of A. Proof Every maximal modular ideal I of A has codimension 1 and it is closed (for the latter, see [8, (B.5.2) Proposition]); thus, A/I is a Banach algebra, which has no proper ideals. The modularity of I implies that A/I has a unit u + I, with u ∈ A. We now prove that each non-zero element of A/I has an inverse. Indeed, if b + I = 0, then M = {ab + I : a ∈ A} is a non-zero ideal (it contains b + I) in A/J ; hence, M = A/I. Thus, u + I ∈ M, so that there exists a0 ∈ A, such that a0 b + I = u + I; but, a0 b + I = (a0 + I)(b + I) and so a0 + I is the inverse of b + I. Now Theorem A.3.16, implies that M is isometrically isomorphic to C. The quotient map : A → A/I, whose kernel is I, is an algebra homomorphism with (u) = u + I = 0, therefore it is a character of A. As for the uniqueness, if there were two characters ω and ω , with kernel I, since u and I generate A (Proposition A.6.1) and ω(u) = ω (u) = 1, they necessarily coincide.   The following theorem allows to describe the spectrum of an element in terms of characters. Theorem A.6.5 Let A be a commutative Banach algebra with unit e. Let a ∈ A. Then, λ ∈ σA (a), if and only if, there exists ω ∈  A, such that ω(a) = λ. Proof If λ ∈ σA (a), the element a − λe is not invertible. This implies that I = {(a − λe)x : x ∈ A} is a proper ideal of A containing a − λe. This ideal is contained in a maximal ideal J . This ideal is clearly modular, since e + J is the unit of A/J . By Proposition A.6.4, J is the kernel of a character ω. Then, ω(a − λe) = 0; thus ω(a) = λω(e) = λ. Conversely, if λ ∈ σA (a), then ω(a − λe) = 0 for every ω ∈  A, otherwise, we would have

1 = ω(e) = ω (a − λe)(a − λe)−1 ) = ω(a − λe)ω (a − λe)−1 = 0, a contradiction.

 

Before going forth, we recall some facts of duality theory that will be needed in what follows.

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If A is a Banach space, as we have already seen, its topological dual space A , i.e., the space of all bounded linear functionals on A, is a Banach space under the norm ω = sup |ω(a)|, a≤1

ω ∈ A .

The unit ball UA is not compact for the norm topology (unless A is finitedimensional). There is, however, another natural topology that can be defined on A , namely the weak* topology, which is a locally convex topology defined by the seminorms A # ω → |ω(a)|,

a ∈ A,

where according to the celebrated Banach–Alaoglou theorem, UA is weakly*compact in A . But, UA is also convex; then, the Krein–Milman theorem states that UA has extreme points i.e., elements that cannot be expressed as convex combinations of two other elements of UA and that UA is the weak*-closure of the set of all combinations of its extreme points. One of the consequences of this fact is that for the Gelfand–Naimark seminorm introduced in Sect. A.5, one has that p(a) = sup ω(a ∗ a)1/2 = ω∈S (A)

sup

ω∈E S (A)

ω(a ∗ a)1/2

(A.6.1)

where ES(A) is the set of extreme points of S(A). Theorem A.6.6 If A is a commutative Banach algebra with unit, then the space  A of characters of A is weakly*-compact in A . Proof By Theorem A.6.2,  A ⊂ UA . It suffices to show that  A is weakly*-closed  in UA First, we notice (but we omit the easy proof) that the limit ω of a net {ωγ } of characters is still a multiplicative linear functional. There are two possibilities: either ω = 0 or ω ∈  A. The case ω = 0 is excluded by the fact that A has a unit and for each γ , ωγ (e) = 1.   We are now ready to define the Gelfand transform. Let A be a commutative Banach algebra with unit e and  A the space of characters of A. For a ∈ A we define a complex valued function  a on  A, by  a (ω) := ω(a), for ω ∈  A. The function  a is continuous on  A endowed with the weak* topology. Indeed, one has | a (ω) −  a (ω )| = |ω(a) − ω (a)|, where on the right hand side, seminorms defining the weak* topology of A appear. Therefore,  a ∈ C( A), the Banach *-algebra of continuous functions on  A, whose supremum norm will be denoted as  · ∞ . The function  a is called the Gelfand

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transform of a and the homomorphism A → C( A) : a →  a is called Gelfand map. Theorem A.6.7 (Gelfand) Let A be a commutative Banach algebra with unit e. The Gelfand map is a homomorphism of A into C( A), with the property  a ∞ ≤ a, for all a ∈ A. Hence, the preceding map is continuous and moreover,  a ∞ = |a|σ = ν(a),

∀ a ∈ A.

Proof By the very definitions one has that  a ∞ = sup |ω(a)| ≤ sup ω a ≤ a,  ω∈A

 ω∈A

∀ a ∈ A.

Moreover, by Theorem A.6.5,    a ∞ = sup |ω(a)| = sup |λ| : λ ∈ σA (a) = |a|σ ,  ω∈A

∀ a ∈ A.

 

Remark A.6.8 We notice that, if A is a commutative Banach *-algebra with unit, the Gelfand map is not, in general, *-preserving (see Definition A.6.12(1) below), unless each character is hermitian, like in the case of commutative C*-algebras. In this situation one has ∗ (ω) = ω(a ∗ ) = ω(a) =  a (ω), a"

a ∈ A.

The hermiticity of a character in the aforementioned case follows from the fact that the spectrum of a hermitian element in an arbitrary C*-algebra is real [8, (8.1) ∗ = Proposition]). Therefore, a" a , for each a ∈ A. The kernel of the Gelfand map     a∈A: a = 0 = a ∈ A : ν(a) = 0 coincides, according to Propositions A.6.1 and A.6.4, with the intersection of all maximal modular ideals of A, i.e., the (Jacobson) radical of A. This set can be proved to be equal to the ‘radical’ of the algebra A, defined as the intersection of the kernels of all characters of A. Thus, if A is semisimple, the Gelfand map is injective. We now consider the case where A is a C*-algebra. If A is a Banach *-algebra with unit, taking into account the definitions (see discussion before Proposition A.5.3), the radical of A is contained in the *-radical of A (cf., e.g., [8, (30.2) Proposition]). Thus, if A is *-semisimple, it is also semisimple and, if A

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is commutative, the Gelfand map is injective. By Theorem A.5.5 any C*-algebra is *-semisimple. Theorem A.6.9 Let A be a commutative C*-algebra with unit. Then, the Gelfand map a →  a is an isometric *-isomorphism of A onto the C*-algebra C( A) of continuous functions on the weakly*-compact space  A of characters of A. We need the following lemma (see, e.g., [8, (29.5) Theorem]) Lemma A.6.10 Let A be a commutative C*-algebra with unit e and ω a positive linear functional with ω(e) = 1, i.e., ω ∈ S(A). Then, the following statements are equivalent: (1) ω is an extreme point of S(A); (2) ω is a character. Proof of Theorem A.6.9 We prove that the Gelfand map a →  a is isometric. Indeed, using Theorem A.5.5 and (A.6.1), we have  a 2∞ = sup | a (ω)|2 = sup |ω(a)|2  ω∈A

 ω∈A

= sup ω(a ∗ )ω(a) = sup ω(a ∗ a) =  ω∈A

 ω∈A

sup

ω∈E S (A)

ω(a ∗ a) = a2 .

It remains only to prove that the Gelfand map is surjective. Let X denote the image of A under the map a →  a . Then, X is norm-closed in C( A) and contains the unit function. Clearly, for any pair ω1 , ω2 ∈  A, such that ω1 = ω2 , there exists a ∈ A with ω1 (a) = ω2 (a) or equivalently,  a (ω1 ) =  a (ω2 ). So X separates the points of  A. Thus, by the Stone–Weierstrass theorem, it follows that X = C( A).   With the help of the Gelfand map, some very important properties of non commutative C*-algebras, can be showed. As an instance, we prove the following Proposition A.6.11 Let A be a C*-algebra with unit and a = a ∗ in A. Then, there exist two positive elements a+ , a− ∈ A, such that a+ a− = a− a+ = 0 and a = a+ − a−

Proof Let M(a) denote the commutative unital C*-subalgebra of A generated by a. Then, M(a) is isometrically isomorphic to C(X), for some compact Hausdorff space X. Let  a be the Gelfand transform of a and  a+ ,  a− the respective positive and negative parts of  a in C(X). Taking the inverse images of these functions, we find the requested elements of A.  

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A.6.2 The Noncommutative Case Definition A.6.12 Let A, B be *-algebras. A map τ *-homomorphism if (1) τ (a ∗ ) = τ (a)∗ , for all a ∈ A;

:

A

→

B is a

(2) τ (αa + βb) = ατ (a) + βτ (b), for all α, β ∈ C and a, b ∈ A; (3) τ (ab) = τ (a)τ (b), for all a, b ∈ A. If A and B have units, denoted by eA , eB respectively, we may suppose without loss of generality that τ preserves units; i.e., τ (eA ) = eB . If it is not so we replace B with τ (eA )Bτ (eA ). Proposition A.6.13 The following hold: (1) if A, B are Banach algebras with units and τ a homomorphism between them, then σB (τ (a)) ⊆ σA (a), for all a ∈ A; (2) if A, B are Banach *-algebras with units and τ a *-homomorphism between them, then for a ∈ A, positive (i.e., a ≥ 0), one has that τ (a) is positive in B, too; (3) if A, B are C*-algebras and τ a *-homomorphism between them, then τ is continuous and, in particular, τ (a) ≤ a, for all a ∈ A. Proof 1. Let λ ∈ ρA (a). Then, (a − λeA )−1 exists in A. This implies that also τ (a) − λeB has an inverse in B, since

(τ (a) − λeB )τ (a − λeA )−1 = τ (a − λeA )τ (a − λeA )−1 = τ (eA ) = eB .

Thus, λ ∈ ρB τ (a) . 2. Since a ≥ 0, we have that a ∗ = a and σA (a) ⊂ R+ ∪ {0} (cf. Definition A.3.23), therefore τ (a)∗ = τ (a ∗ ) = τ (a), for all a ∈ A. The assertion now follows from (1). 3. Add units eA , eB , in A and B respectively, if necessary and suppose that τ (eA ) = eB . Then, for any a ∈ A, we have τ (a)2 = τ (a)∗ τ (a) = τ (a ∗ a) = ν(τ (a ∗ a)). Now, by (1) and Corollary A.3.5, it follows that ν(τ (a ∗ a)) ≤ ν(a ∗ a) = a ∗ a = a2 and this completes the proof of (3).   Corollary A.6.14 Each (algebraic) *-isomorphism of C*-algebras is isometric. The foregoing statements apply in particular to *-representations of a C*-algebra that are, by definition, *-homomorphisms of A into B(H). In particular, any faithful *-representation π (i.e., π(a) = 0 implies a = 0) is norm-preserving and π(A) is a C*-algebra of bounded operators. Before proving the main theorem of this section, we need the notion of direct sum of *-representations.

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Let A be a C*-algebra. For each j ∈ I , where I is a set of indices, let πj be a *-representation of A on the Hilbert space Hj . First, we construct the direct sum 

H=

Hj

j ∈I

of the Hilbert spaces Hj in the following way. The finite subsets F of I form a directed set with the set inclusion as order. An element ξ of H is a family ξ = {ξj } with ξj ∈ Hj , such that lim F



ξj 2j < ∞,

j ∈F

where  · j denotes the norm in Hj derived from the inner product ·|·j , j ∈ J . The inner product in H is then defined by ξ |η = lim F



ξj |ηj j , ξ, η ∈ H.

j ∈F

The direct sum *-representation π ≡

.

j ∈I

    π(a) ξj = πj (a)ξj ,

πj is given by   a ∈ A, ξj ∈ H.

Since πj (a) ≤ a, j ∈ J , a ∈ A, we have that supj ∈J πj (a) < ∞. Moreover, π(a) = supj ∈J πj (a) ≤ a, therefore π(a) is a bounded operator on H, for all a ∈ A. Theorem A.6.15 (Gelfand–Naimark) Each C*-algebra A with unit has a faithful *-representation. It follows that A is isometrically *-isomorphic to a C*-algebra of bounded linear operators on a Hilbert space. Proof For each ω ∈ S(A), the GNS construction yields a bounded *representation πω of A on a Hilbert space Hω (see Theorem A.2.16). Let π be the *-representation direct sum of the family {πω }ω∈S (A) , acting on the Hilbert space . H = H ω∈S (A) ω . By Proposition A.5.4, for each non-zero a ∈ A there exists ω ∈ S(A), such that πω (a) = a. Then, a = πω (a) ≤ π(a). By (3) of Proposition A.6.13, we finally get π(a) = a, a ∈ A, thus π is a faithful *-representation. In conclusion, A is isometrically *-isomorphic to the C*-algebra   π(A). Remark A.6.16 The *-representation π constructed in the proof of Theorem A.6.15 is usually called universal *-representation of A . Thus, one can formulate Theorem A.6.15 by saying that π is an isometric *-isomorphism of A onto a C*algebra of bounded linear operators on the carrier space H.

Appendix B

Operators in Hilbert Spaces

In this chapter after recalling shortly some basic facts on Hilbert spaces, we list, mainly without proofs, the fundamental aspects of the theory of bounded and unbounded operators in Hilbert spaces that have been used throughout this book.

B.1 Hilbert Spaces Definition B.1.1 Let D be a complex vector space. A map associating to an ordered pair (ξ, η) of elements of D×D the complex number ξ |η is called an inner product if (i) (ii) (iii) (iv)

αξ + βη|ζ  = αξ |ζ  + βη|ζ ; ξ |η = η|ξ ; ξ |ξ  ≥ 0; ξ |ξ  = 0 ⇔ ξ = 0;

where ξ, η, ζ ∈ D and α, β ∈ C. A vector space D endowed with an inner product is called an inner product space or pre-Hilbert space Now, put ξ  := ξ |ξ 1/2 ,

ξ ∈ D;

(B.1.1)

then, it is easily seen that the triangle inequality ξ + η ≤ ξ  + η,

∀ ξ, η ∈ D,

© Springer Nature Switzerland AG 2020 M. Fragoulopoulou, C. Trapani, Locally Convex Quasi *-Algebras and their Representations, Lecture Notes in Mathematics 2257, https://doi.org/10.1007/978-3-030-37705-2

237

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holds true. This fact, together with (iii) and (iv) of Definition B.1.1, implies that (B.1.1) actually defines a norm on D. Moreover, the following Cauchy–Schwarz inequality holds |ξ |η| ≤ ξ η,

∀ ξ, η ∈ D.

(B.1.2)

Therefore, every pre-Hilbert space is a normed space. Remark B.1.2 From the Cauchy–Schwarz inequality it follows easily that ξ  = sup |ξ |η|. η≤1

(B.1.3)

Definition B.1.3 Two vectors, ξ and η of D are said to be orthogonal if ξ |η = 0. A set ξi of vectors of D is called orthonormal if ξi |ξi  = 1 and ξi |ξj  = 0, for i = j . Remark B.1.4 For two orthogonal vectors ξ and η of a pre-Hilbert space D the Pythagoras theorem holds; i.e., ξ + η2 = ξ 2 + η2 . Definition B.1.5 A pre-Hilbert space H which is complete with respect to the norm (B.1.1) is said to be a Hilbert space.   If M ⊂ H, then M⊥ ≡ ξ ∈ H : ξ |η = 0, ∀ η ∈ M is a closed subspace of H. If M itself is a closed subspace of H, then M⊥ is called the orthogonal complement of M. It can be proved that, if M is a closed subspace of H, then M ⊕ M⊥ = H. Similarly, M is dense in H, if and only if, M⊥ = {0} Example B.1.6 (a) For n ∈ N fixed, the space Cn of all n-tuples of complex numbers z = (z1 , z2 , . . . , zn ) is a Hilbert space if the inner product of z and w = (w1 , w2 , . . . , wn ) is defined by z|w =

n 

zj w¯ j .

j =1

(b) The space L2 (R) of Lebesgue square integrable functions (modulo the set of all almost everywhere null functions) is a Hilbert space if the inner product of two elements f, g in L2 (R) is defined by  f |g =

R

f (x)g(x) dx, x ∈ R.

(B.1.4)

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We notice that (B.1.4) is well defined because of Hölder inequality. The completeness of L2 (R) constitutes the content of the Riesz–Fisher theorem. (c) The space C[0, 1] of all complex valued continuous functions in [0, 1] is an inner product space if the inner product is defined by 

1

f |g =

f (x)g(x) dx, x ∈ [0, 1],

0

but it is not a Hilbert space. Indeed, let us consider the sequence of functions

fn (x) =

⎧ ⎪ ⎨  n

2 ⎪ ⎩

0, 

x−

if 0 ≤ x ≤

1 2

+ 12 , if

1,

if

1 2 1 2

− +

1 n 1 n

1 2

−

≤ x≤

1 n 1 2

+

1 n

≤ x≤ 1

for n > 2. It is easy to see that if f is the discontinuous function  f (x) =

0, if 0 ≤ x ≤ 12 1, if 12 < x ≤ 1

one has  lim

n→∞ 0

1

|fn (x) − f (x)|2 dx = 0, x ∈ [0, 1].

Thus (fn ) is a Cauchy sequence in C[0, 1] but f ∈ C[0, 1]. Let E[ · E ], F [ · F ] be normed spaces. We remind that a linear map f : E → F , i.e., f (αx + βy) = αf (x) + βf (y),

∀ x, y ∈ E, α, β ∈ C,

is said to be bounded if there exists a constant C > 0, such that f (x)F ≤ CxE ,

∀ x ∈ E.

It is well known that boundedness of f is equivalent to its continuity at 0 (and then at every point of E). The norm of a continuous linear map f : E[·E ] → F [·F ] is defined as   f  := sup f (x)F : xE ≤ 1 , x ∈ E. In the case F = C, a linear map from E into C is called a linear functional on E. The space of all bounded (continuous) linear functionals on E is denoted by E  and

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called the dual space of E. The space E  is a Banach space under the norm   f  := sup |f (x)| : xE ≤ 1 , f ∈ E  , x ∈ E. The following theorem, known as Riesz Lemma, is one of the main results of the theory of Hilbert spaces. It is due to Riesz and Fréchet and characterizes bounded linear functionals on a Hilbert space H. Theorem B.1.7 Let H be a Hilbert space and η ∈ H. Let fη (ξ ) = ξ |η, ξ ∈ H. Then, fη is a continuous linear functional on H and fη  = η. Conversely, if f is a continuous linear functional on H, then there exists a unique η ∈ H, such that f = fη . Proof The fact that fη is continuous, follows immediately by (B.1.2). The same inequality shows that fη  ≤ η. On the other hand, 

      fη  = sup |fη (ξ )| : ξ  = 1 ≥ fη η−1 η  = η−1 η|η = η and this concludes the proof of first part. Conversely, let f be a continuous linear functional on H. Put M = Kerf ; then, M is a closed vector subspace of H, which does not coincide with H. Then, M⊥ = {0}. Taking u ∈ M⊥ with u = 1, we have f (u)f (ξ ) − f (ξ )f (u) = 0, therefore f (u)ξ − f (ξ )u ∈ M. Since u ∈ M⊥ , we obtain 0 = f (u)ξ − f (ξ )u|u = f (u)ξ |u − f (ξ ), that is f (ξ ) = f (u)ξ |u. If we put η = uf (u), we obtain f = fη . As for the uniqueness, let ζ ∈ H be another vector in H, such that f = fζ . Then, η − ζ  = fη−ζ  = fη − fζ  = f − f  = 0, whence η = ζ .

 

B.2 Bounded Operators In the discussion before Theorem B.1.7, the definition of a bounded linear map between Banach spaces and its norm are given. Since every Hilbert space H is a Banach space, a bounded linear map T : H → H is called a bounded operator

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241

and its norm T  is exactly the norm of a bounded linear map T between Banach spaces; i.e.,   T  := sup T ξ  : ξ  ≤ 1 , ξ ∈ H. The set of all bounded operators on a Hilbert space H will be denoted by B(H). Remark B.2.1 It is easy to prove that the sum, scalar multiple and product of bounded operators are again bounded operators, such that T + S ≤ T  + S,

αT  = |α|T ,

T S ≤ T  S,

for any T , S ∈ B(H) and α ∈ C. An interesting application of Riesz lemma is the following Theorem B.2.2 Let ( , ) be a bounded sesquilinear form on H, i.e., a map from H × H into C satisfying the following conditions: (i) (αξ + βη, ζ ) = α(ξ, ζ ) + β(η, ζ ); (ii) (ξ, αη + βζ ) = α(ξ, η) + β(ξ, ζ ); (iii) there exists a constant C > 0, such that |(ξ, η)| ≤ Cξ η, for any ξ, η, z ∈ H α, β ∈ C. Then, there exists a unique bounded linear operator T from H into H, such that (ξ, η) = ξ |T η,

∀ ξ, η ∈ H

and   T  = sup |(ξ, η)| : ξ  = η = 1 .

(B.2.1)

Proof Fix η ∈ H; then η (ξ ) = (ξ, η) is a bounded linear functional on H. By Riesz lemma, there exists ζ ∈ H, such that η (ξ ) = (ξ, η) = ξ |ζ ,

∀ ξ ∈ H.

Put Aη = ζ . One defines in this way a map A from H into itself. It is easy to prove that A is a linear operator. To see that it is bounded, we compute Aη2 , for η ∈ H; indeed, Aη2 = Aη|Aη = (Aη, η) ≤ CAη η. It remains to prove uniqueness. Let A be another linear operator, such that (ξ, η) = ξ |Aη, for all ξ, η ∈ H. Then, ξ |A η − Aη = 0, for all ξ ∈ H; but H⊥ = {0}. The equality (B.2.1) follows easily from (B.1.3).  

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B.3 The Adjoint of a Bounded Operator Theorem B.3.1 Let T ∈ B(H). Then, there exists an operator T ∗ ∈ B(H) (called the adjoint of T ), such that T ξ |η = ξ |T ∗ η,

∀ ξ, η ∈ H.

Proof For fixed η ∈ H, put ωT ,η (ξ ) ≡ T ξ |η,

ξ ∈ H.

Then, ωT ,η is a linear functional on H and since |ωT ,η (ξ )| = |T ξ |η| ≤ T ξ η,

ξ ∈ H,

it is bounded. By the Riesz lemma, there exists ζ ∈ H, such that ωT ,η (ξ ) = T ξ |η = ξ |ζ ,

ξ ∈ H.

Now put T ∗ η = ζ . The map defined in this way is linear. We prove that it is bounded. We have T ∗ η2 = T ∗ η|T ∗ η = T T ∗ η|η ≤ T  T ∗ η η. This proves that T ∗ ∈ B(H) and also that T ∗  ≤ T . Proposition B.3.2 The map T → T ∗ in B(H) has the following properties:

 

(1) T ∗∗ = T , for every T ∈ B(H); (2) (αT + βS)∗ = αA∗ + βS, for all T , S ∈ B(H) and α, β ∈ C; (3) (T S)∗ = S ∗ T ∗ , for all T , S ∈ B(H). Remark B.3.3 From (1) of Proposition B.3.2 and the last part of the proof of Theorem B.3.1 it follows easily that T ∗  = T , for every T ∈ B(H). Before going forth, we define some special class of bounded operators.

B.3.1 Symmetric Operators A bounded operator T in H is called selfadjoint if T = T ∗ , i.e., if T ξ |η = ξ |T η,

∀ ξ, η ∈ H.

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As an example, we consider H = L2 (0, 1) and the operator Q : L2 (0, 1) → L2 (0, 1), such that (Qf )(x) := xf (x), for every f ∈ L2 (0, 1) and x ∈ (0, 1). It is readily seen that Q is bounded and selfadjoint. For any selfadjoint operator T and any ξ ∈ H, the number T ξ |ξ  is real. A selfadjoint operator T is said to be positive, and we write T ≥ 0, if T ξ |ξ  ≥ 0, for every ξ ∈ H.

B.3.2 Projection Operators A bounded operator P is called a projection operator if P = P 2 = P ∗ . There is one-to-one correspondence between closed subspaces of H and projection operators on H. As we already remarked, if M is a closed subspace of H, then H = M⊕M⊥ . If P denotes the canonical projection of H onto M, then it turns out that P is a projection operator on H. Conversely, if P is a projection operator, then the set  MP = ξ ∈ H : P ξ = ξ is a closed subspace of H and P is the projection operator on MP .

B.3.3 Isometric and Unitary Operators A linear operator U : H → H is called isometric if U ξ |U η = ξ |η, for every ξ, η ∈ H. One obviously has U ξ  = ξ , for every ξ ∈ H; therefore, U is bounded and U  = 1. The operator U is injective and so U −1 exists. If U −1 = U ∗ , the operator U is called unitary. Equivalently, U is unitary if U ∗ U = U U ∗ = I , where I denotes the identity operator of H. For a fixed t ∈ R, the shift operator U : L2 (R) → L2 (R), such that Uf (t) := f (x + t), x ∈ R, is an example of a unitary operator.

B.4 Unbounded Operators in Hilbert Spaces Definition B.4.1 Let H be a Hilbert space, D(T ) a dense vector subspace of H and T : D(T ) → H a linear map, i.e., T (αξ + βη) = αT ξ + βT η,

∀ ξ, η ∈ D(T ) and ∀α, β ∈ C.

Then, the pair (T , D(T )) is called a linear operator in H. We refer to D(T ) as to the domain of T .

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The set   Ran (T ) = η ∈ H : η = T ξ, for some ξ ∈ D(T ) will be called the range of T . We will often speak of the operator T , without writing explicitly its domain. The specification of the domain is, however, essential for an appropriate definition of the operator. Definition B.4.2 An operator (T  , D(T  )) is called an extension of the operator (T , D(T )) and we write T ⊂ T  if D(T ) ⊂ D(T  ) and T η = T  η, for every η ∈ D(T ). The algebraic operations between linear operators T and S are defined as follows (provided that the sets indicated below as domains are dense in H): (i) Addition T + S:  D(T + S) = D(T ) ∩ D(S), (T + S)ξ = T ξ + Sξ, ξ ∈ D(T + S). (ii) The multiplication λT of T by scalars λ ∈ C: If  λ = 0, then λT ≡ 0, otherwise D(λT ) = D(T ), (λT )ξ = λ(T ξ ), ξ ∈ D(T ). (iii) Themultiplication T S:   D(T S) = ξ ∈ D(S) : Sξ ∈ D(T ) , (T S)ξ = T (Sξ ), ξ ∈ D(T S). (iv) The inverse T −1 : If  T is injective, then D(T −1 ) = R(T ), T −1 (T ξ ) = ξ,

ξ ∈ D(T ).

The usual associative laws hold for the addition and multiplication. That is, given the operators T , S, Q in H, one has (T + S) + Q = T + (S + Q) and (T S)Q = T (SQ). The distributive law holds too, but one has the inclusion T (S + Q) ⊃ T S + T Q instead of equality. Let T be a linear operator in H. The set   G(T ) ≡ (ξ, T ξ ) : ξ ∈ D(T ) is a subspace of the direct sum H ⊕ H, called the graph of T . It is clear that T = S, if and only if, G(T ) = G(S), and T ⊂ S, if and only if, G(T ) ⊂ G(S).

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245

Definition B.4.3 A linear operator T in H is said to be closed if its graph G(A) is closed in H ⊕ H, that is, if {ξn } ⊂ D(T ) is such that ξn → ξ and T ξn → η, then ξ ∈ D(T ) and η = T ξ . Let T be a closed operator in H. Then, D(T ) can be made into a Hilbert space under the inner product ξ |ηT := ξ |η + T ξ |T η,

ξ, η ∈ D(T ).

This Hilbert space is denoted by HT and the norm  · T is called the graph norm associated to T . Definition B.4.4 A linear operator T in H is said to be closable if it has a closed extension. Every closable operator T has a minimal closed extension, called its closure and denoted by T . Then, we have the following Proposition B.4.5 Let T be a linear operator in H. The following statements are equivalent: (i) T is closable; (ii) G(T ) is the graph of a linear operator in H; (iii) If {ξn } ⊂ D(T ), such that ξn → 0 and T ξn → η, then η = 0. If one of the preceding conditions is true, then T is given by    D(T ) = ξ ∈ H : ∃ {ξn } ⊂ D(T ), such that ξn → ξ and T ξn → η , T ξ = η,

ξ ∈ D(T ).

If T is closed and T −1 exists, then T −1 is closed. Next we define the adjoint T ∗ of a densely defined linear operator T in H: 

  D(T ∗ ) = η ∈ H : ∃ ζ ∈ H, such that (T ξ |η) = (ξ |ζ ), for all ξ ∈ D(T ) , T ∗ ξ = ζ, ξ ∈ D(T ∗ ).

Since D(T ) is dense in H, T ∗ is a well-defined linear operator in H but, unlike the case of bounded operators, D(T ∗ ) may not be dense in H. For example, let f0 be a bounded measurable function on R with f0 ∈ L2 (R) and g0 ∈ L2 (R). Then, the densely defined linear operator T in L2 (R) given by 

/   D(T ) = h ∈ L2 (R) : R |h(t)f0 (t)|dt < ∞ , T h = h|f0 g0 ,

has D(T ∗ ) = {0}.

h ∈ D(T )

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The following is immediate. Proposition B.4.6 Let T and S be densely defined linear operators in H. Then, the following statements hold: (1) if T ⊂ S, then S ∗ ⊂ T ∗ ; (2) (λT )∗ = λT ∗ , λ ∈ C; (3) if D(T + S) is dense in H, then

(T + S)∗ ⊃ T ∗ + S ∗ . In particular, if T or S is bounded, then (T + S)∗ = T ∗ + S ∗ ; (4) if D(T S) is dense in H, then (T S)∗ ⊃ S ∗ T ∗ . In particular, if T is bounded, then (T S)∗ = S ∗ T ∗ . The notions of adjoint and closure are intimately related. Theorem B.4.7 Let T be a densely defined linear operator in H. The following statements hold: (1) T ∗ is closed and G(T ∗ ) = V (G(T )⊥ ), where V is the unitary operator on

H ⊕ H defined by V (ξ, η) = −η, ξ , ξ, η ∈ H; (2) T is closable, if and only if, D(T ∗ ) is dense in H. If this is true, then T = T ∗∗ (≡ (T ∗ )∗ ); ∗ (3) if T is closable, then T = T ∗ .

We consider now the particular case of bounded operators.   Theorem B.4.8 If (T , D(T )) is bounded, i.e., sup T ξ  : ξ ∈ D(T ), ξ  ≤ 1 < ∞, then (i) T is closable and T ∈ B(H); (ii) T ∗ is everywhere defined in H and bounded; i.e., T ∗ ∈ B(H). Proof To prove (i) we show that (iii) of Proposition B.4.5 holds. Let {ξn } ⊂ D(T ) be a sequence, such that ξn → 0 and T ξn → η. Since T ξn  ≤ T  ξn ,

∀ n ∈ N,

it follows that (T ξn ) converges to 0. Thus, η = 0 and T is closable. If ξ ∈ H, then there exists a sequence (ξn ) in D(T ), which converges to ξ . Since T is bounded, the

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247

sequence (T ξn ) is Cauchy and thus it converges in H. By the definition of closure itself, it follows then that ξ ∈ D(T ) and T ξ = limn→∞ T ξn . The statement (ii) follows immediately from (i) and from Theorem B.3.1.   Conversely, if T is a closed linear operator on H (that is, defined everywhere), then T ∈ B(H), according to the closed graph theorem given below (Theorem B.4.10). For the proof we need the following Lemma B.4.9 Let {ξn } be a sequence of vectors in a Hilbert space H. Then, the sequence of numbers {ξn |η} is bounded, for every fixed η ∈ H, if and only if, the sequence of norms {ξn } is bounded. Theorem B.4.10 Every closed linear operator T on H (i.e., defined everywhere) is bounded. Proof First we show that T ∗ is continuous. Indeed, take any sequence {ηn } in D(T ∗ ), which converges to 0. Define ηn = ηn −1/2 ηn , n ∈ N, so that ηn  → 0, as well. Then, for any g ∈ H, we have |T ∗ ηn |g| = |ηn |T g ≤ ηn  T g. By Lemma B.4.9, {T ∗ η } is a norm-bounded sequence, which implies that n

lim T ∗ ηn = lim ηn 1/2 T ∗ ηn = 0.

n→∞

n→∞

Thus, T ∗ is continuous. It follows from Theorem B.4.7 that D(T ∗ ) is dense in H and T ∗ is closed, which implies that D(T ∗ ) = H and T ∗ is bounded. Hence,   T = T ∗∗ is also bounded.

B.5 Symmetric and Selfadjoint Operators Definition B.5.1 Let (T , D(T )) be a linear operator in H. We say that T is symmetric if T ξ |η = ξ |T η,

∀ ξ, η ∈ D(T ).

The definition implies that T ⊂ T ∗ and therefore any symmetric operator is closable. Its closure T ∗∗ (see Theorem B.4.7) is also a symmetric operator. Definition B.5.2 A symmetric operator (T , D(T )) is said to be selfadjoint if T = T ∗. Clearly, if T is everywhere defined and symmetric, it is bounded and selfadjoint. Example B.5.3 We produce some examples showing that the notions of symmetric and selfadjoint operators are really different. We define:  /t  D(S) ≡ f ∈ C[0, 1] : f (t) − f (0) = 0 f1 (s) ds, s ∈ [0, 1], for some f1 ∈  L2 [0, 1] , Sf = −if1 , f ∈ D(S)

248

B Operators in Hilbert Spaces

d (S is often denoted by −i dt ). We define some operators in L2 [0, 1], as follows:



  D(T ) = f ∈ D(S) : f (1) = f (0) = 0 , T = S  D(T ),

and, for any α ∈ C,    D(Tα ) = f ∈ D(S) : f (1) = αf (0) , Tα = S  D(Tα ). Then, the following facts hold: 1. T is a symmetric operator, which is not selfadjoint and in fact T ∗ = S. 2. For any α ∈ C, with |α| = 1, the operator 

  D(Tα∗ ) = f ∈ AC[0, 1] : f (0) = αf (1) , T ∗ = S  D(T ∗ ) α

α

is a selfadjoint extension of T . Note that AC[0, 1] is the algebra of all absolutely continuous C-valued functions on [0, 1]. The basic criterion of selfadjointness is provided by the following Theorem B.5.4 A closed symmetric operator T is selfadjoint, if and only if, Ker(T ∗ ± iI ) = {0}.

Bibliography

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Suggested Readings

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Index

(Y , D)s , 17 (Y , D)w , 17 DM (A), 73 Nϕ , 17 X† , 13 ωξ , 25  · B , 35 σα (·), 132 L† (D , H), 13  ·  , 119  ·  , 40  ·  # , 118  · or b , 155  · 1 , 25  · H , 209  · L , 32  · R , 32  ·  , 103  ·  , 118  ·  , 116  · ϕ , 52  · p , 34  · p, , 127  · −1 , 34  · T , 114  · L , 35  · p , 50  · s , 37  · π , 62  · 0 , 32  · # , 102  · b , 68  · lb , 68  · rb , 68

 · +1,−1 , 104  · p,I , 130 (πϕ , λϕ , Hϕ ), 21 (τ, t)-continuous, 145 (A, A0 ), 9 (A[ · ], #), 121 (Aor b )h , 155 B(ϕ), 18 D(ϕ), 145 E S (A), 232 J , 105 JA , 112 L(b), 10, 184 L(A0 ), 106 Lp (), 34 p Lloc (), 195 Lw , 13 La , a ∈ A, 47 Lw (a), a ∈ A, 46 Lw (A), 46 Lx , x ∈ A0 , 31 LN, 10 M(a), 10 MN, 10 N (p), 50 Nω , 23 P ∼ Q, 127 P ≺ Q, 127 Pα , 130 Pϕ , 199 QM, 199 R(a), 10 R w , 13 Ra , a ∈ A, 47

© Springer Nature Switzerland AG 2020 M. Fragoulopoulou, C. Trapani, Locally Convex Quasi *-Algebras and their Representations, Lecture Notes in Mathematics 2257, https://doi.org/10.1007/978-3-030-37705-2

255

256 Rw (a), a ∈ A, 46 Rw (A), 46 Rx , x ∈ A0 , 31 RN, 10 T ⊂ T  , 244 X1 X2 , 13 )(a), 155 ((a), 155 ≈, 207 D [ · 1 ], 34 D × , 12, 34 D × [t × ], 12 Dω , 211 Dπ , 15 Dπ ∗ , 15 Dϕ , 20 Hω , 23, 211 Hπ , 15 Hϕ , 20 ηω , 23 -semisimple, 113 λω , 23, 211 λϕ , 18 λϕ (a), 18 λϕ (A) = A/Nϕ , 18 ·|·A , 73 H+1 , 103 C (W ), 183 F (W )+ , 183 U (A+ 0 ), 164 s, 49 EMT (A0 ), 137 ENT (A0 ), 138 MT (A0 ), 136 NT (A0 ), 138 NTAτ (A) , 198 0 Proj(M), 127 Repcr (A), 49 Repsc (A), 62 rad(·), 230 R(A), 229 RA0 (A), 41 q(·), 157 tu (B), 170 tu∗ (B), 170 ts , 14 tu (B), 170 tw , 14 ts ∗ , 14 μ(A), 128 ωe , 210 ωb , 214 ωx , 58

Index ωϕx , 28 ωϕ , 26 ⊕α∈I πα , 16 La , 67 R a , 68  · , 62  · A , 73  · E , 79  · B , 35 π ∗ , 15 πϕ◦ , 20 ρ(·), 83 ρA , 221 σA , 221 σA0 (·), 186 , 35 τp , 195 τs , 46 τw , 46 τs ∗ , 46 B(D, D), 25 B† (D), 25 #, 105 #-semisimple, 113 , 105 ω , 55 ϕ , 26

, 184 , 199 ϕA , 25 ϕξ , 27 ϕx , 28 a , 47 L a , 47 R  A, 230  ωϕ , 65  π , 15  τp , 195 M 0 [τ ]+ , 163 A 0 [τ ]+ A c , 163 {pξ∗ }, 14 {pξ }, 14 {pξ,η }, 14 {A, ∗, A# , #}, 102 {A, ∗, A , }, 103 ab = a · b, 11 a · b, 10 a b, 69 a b, 68 a • b, 71 a ≥ 0, 147 a b, 46, 158 a −1 , 82

Index

257

e ≡ (0, 1), unit of Ae , 202 m*-seminorm, 48 nth-root, 188 commutatively, 193 p-bounded, 216 qj (·), 195 qm-admissible wedge, 147 r(·), 84 t × = β(D× , D), 12 tπ , 205 x ≤ y, 147 x  , 103 x+ , 175 x− , 175 I, 13 IDπ , 15 B(H), 241 B(A), 68 F , 149 s (A), 29 IA 0 IA0 (A), 18 Jp , 33, 126 L, 77 L† (D ), 13 L† (D )b , 14 L† (D , H)b , 13 Mc , 77 Mξ , 26 QA0 (A), 17 R, 77 R(A, A0 ), 22 Rc (A, A0 ), 54 Ruc (A, A0 ), 56 S (R), 11 S (A), 228 S  (R), 11 τ SA (A), 194 0 SA0 (A), 39 TAτ 0 (A), 194 T (A), 113 T# (A), 113 TA0 (A), 135 U (E), 79 UA , 43 p B+ , 129 PA0 (A), 39 ·

, 42 A+ := A+ 0 τ , 147 A+ := A+ 0 0 [τ ]+ , 164 A+ := A ∩ A + Ac , 164 A , 39 AP , 43 AS , 43

A+ , 43 Ae , 201 Aor b , 155 Ah , 147, 227 AS , 90 Aq , 88 Ab , 68 Alb , 67 Arb , 67 Asb , 75 G, 219 Iω , 211 L(D), 12 L(D× ), 12 L† (D), 25 M(A0 , A+ ), 175 M†S , 35 Mα , 131 P(A0 ), 190 Sb (A), 83 p, 40 q, 49 qI , 37 M(p, I ) ≡ Mp , 131 Rep(A), 29 Repr (A), 29  · rb , 68 πω , 23 ∗ X1 † , 13 + A0 , 42, 147, 163 p A0 , 50 A0 , 106 A0 , 106 A# , 102 L(D, D× ), 12 L† (D), 13 *-algebra, 201 Banach, 202 normed, 202 partial, 10 *-bimorphism, 121 *-homomorphism, 14, 117 contractive, 117 *-isomorphism, 117 isometric, 117 *-representation, 15 adjoint of, 15 algebraic direct sum, 16 bounded, 16 closed, 15 closure of, 15 completely regular, 49 cyclic, 16, 208 direct sum, 16, 236

258 faithful, 235 irreducible, 17 regular, 27 selfadjoint, 16 strongly continuous, 41 strongly* continuous, 41 strongly-cyclic, 16, 208 ultra–cyclic, 16 universal, 93, 236 weakly continuous, 41 *-subalgebra quasi, 14 N0 (p), 50 σ (·), 83  ωϕ , 65 {A, ∗, A# , #}, 102 A , 102  · p , 65 D(X), 13 adjoint of an operator, 242 admissible subset, 170 algebra Banach, 202 maximal Tomita, 106 normed, 202 semisimple, 230 Banach *-algebra *-radical, 229 *-semisimple, 229 Banach quasi *-algebra full, 73, 168 fully closable, 80 normal, 72 quasi regular, 86 regular, 86 strongly regular, 138 bicommutant, 204 bounded element left, 67 order, 155 right, 67 strongly, 75 bounded inverse, 82 bounded linear map, 239 bounded part, 13 BQ*-algebra, 33 C*-algebra, 202 partial, 159

Index Calkin algebra, 204 C*-condition, 202 character, 230 commutant (von Neumann), 204 double, 204 strong bounded, 17 weak bounded, 17 condition (P ), 150 condition (P), 149 core for ϕ, 19 CQ*-algebra, 102 *-semisimple, 120 proper, 33, 95 pseudo, 111 strict, 111 C*-seminorm extended, 88 unbounded, 159 EC*-algebra, 174 element closable, 80 hermitian, 175 GB*-algebra, 167 Gelfand map, 233 Gelfand–Naimark–Segal (GNS) construction, 21, 212 representation, 21 Gelfand–Naimark seminorm, 229 Gelfand transform, 233 graph norm, 245

HCQ*-algebra, 121 extension, 123 standard, 123 Hilbert algebra, 34 full, 73 left, 105 bicommutant, 106 commutant, 106 full or achieved, 106 right, 106 Hilbert space direct sum, 16 Hilbertian norm, 34 homomorphism, 14

ideal, 230 maximal, 230 modular, 230

Index integral, 128 invariant subspace, 204 invertible element, 219 involution, 201 isometric, 202 ips-form, 18 left-multiplier, 184 linear functional, 239 hermitian, 209, 226 positive, 43, 148, 209 admissible, 215 extensible, 212 faithful, 209 Hilbert bounded, 209 state, 209 representable, 22 continuous, 145 uniformly representable, 56 locally convex quasi C*-algebra, 167 *-semisimple, 194 strongly *-semisimple, 198 quasi C*-normed algebra, 144 O*-algebra, 35 operator adjoint, 12 bounded, 241 closable, 13, 245 closed, 245 closure of an, 13, 245 domain of an, 243 extension of an, 244 graph of an, 244 isometric, 243 -measurable, 127 measurable, 127 modular, 105 modular conjugation, 105 positive, 243 projection, 243 range of an, 244 selfadjoint, 242, 247 symmetric, 247 unitary, 243 orthogonal complement, 238 orthogonal vectors, 238 orthonormal set, 238 partial *-algebra abelian, 10

259 associative, 10 semi-associative, 11 partial multiplication weak, 13 positive element, 42, 163, 224 commutatively, 163 projection -finite, 127 finite, 127 purely infinite, 127

quasi *-algebra, 9 Banach, 31 Hilbert, 34 Hilbertian, 34 locally convex, 143 fully representable, 150 normed, 31 *-radical, 41 *-semisimple, 42 unital, 9 quasi-unit, 9 qu*-homomorphism, 15 qu-homomorphism, 15 qu*-ideal, 15 qu-ideal, 15 left, 15 right, 15 qu*-representation, 15

radical Jacobson, 230 relatively p-bounded, 216 representation, 204 algebraically irreducible, 204 bounded, 204 GNS, 211 ultracyclic, 204 resolvent function, 83 set, 83 resolvent set, in an algebra, 221 rigged Hilbert space, 12 right multiplier weak, 13

sesquilinear form admissible, 21 s-bounded, 36 bounded, 37, 241 closable, 52

260 closed, 52 faithful, 124 invariant, 190 positive, 17 closable, 145 A0 -singular, 19 space Gelfand, 175 Hilbert, 238 inner product, 237 pre-Hilbert, 237 spectral radius, 84, 223 spectrum, 83, 186, 221 of an element in A+ , 186 square-root, 188 commutatively, 193 sub *-representation, 204 subspace closed invariant, 16 invariant, 16 reducing, 16 strongly -dense, 127 strongly dense, 127 sufficient family, 40 of normal finite traces, 130 of positive linear functionals, 149 of strongly continuous *-representations, 66 sufficient set, 66

Index topology B-uniform, 170 graph, 205 strong, 14 strong*, 14 strong dual, 12 strongly B-uniform, 170 strongly∗ B-uniform, 170 weak, 14

unit, 9, 201 unitarily equivalent, 207 unitization, 10, 201

vector cyclic, 208 form, 27 strongly-cyclic, 208 von Neumann algebra finite, 127 purely infinite, 127

weak multiplication, 46 weak multiplication , 158 wedge, 183

LECTURE NOTES IN MATHEMATICS

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