Tomita's Lectures on Observable Algebras in Hilbert Space (Lecture Notes in Mathematics) 3030688925, 9783030688929

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Lecture Notes in Mathematics  2285

Atsushi Inoue

Tomita's Lectures on Observable Algebras in Hilbert Space

Lecture Notes in Mathematics Volume 2285

Editors-in-Chief Jean-Michel Morel, CMLA, ENS, Cachan, France Bernard Teissier, IMJ-PRG, Paris, France Series Editors Karin Baur, University of Leeds, Leeds, UK Michel Brion, UGA, Grenoble, France Camillo De Lellis, IAS, Princeton, NJ, USA Alessio Figalli, ETH Zurich, Zurich, Switzerland Annette Huber, Albert Ludwig University, Freiburg, Germany Davar Khoshnevisan, The University of Utah, Salt Lake City, UT, USA Ioannis Kontoyiannis, University of Cambridge, Cambridge, UK Angela Kunoth, University of Cologne, Cologne, Germany Ariane Mézard, IMJ-PRG, Paris, France Mark Podolskij, University of Luxembourg, Esch-sur-Alzette, Luxembourg Sylvia Serfaty, NYU Courant, New York, NY, USA Gabriele Vezzosi, UniFI, Florence, Italy Anna Wienhard, Ruprecht Karl University, Heidelberg, Germany

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Atsushi Inoue

Tomita’s Lectures on Observable Algebras in Hilbert Space

Atsushi Inoue Department of Applied Mathematics Fukuoka University Fukuoka, Japan

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-030-68892-9 ISBN 978-3-030-68893-6 (eBook) https://doi.org/10.1007/978-3-030-68893-6 Mathematics Subject Classification: Primary: 46L99; Secondary: 46K10 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed 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

Dedicated to the memory of my former supervisor late Professor Minoru Tomita

Preface

In 1967, M. Tomita presented his research on “standard forms of von Neumann algebras” at the international conference on C ∗ -algebras and their physical applications held at the University of Louisiana and at the fifth functional analysis symposium of Mathematical Society of Japan held in Sendai. This is an original theory that is the essence of noncommutative analysis. In 1970, M. Takesaki developed this theory and published the outcome of his work under the title “Tomita’s Theory of Modular Hilbert Algebras and Its Applications” in Springer’s Lecture Notes in Mathematics. This is known as the Tomita-Takesaki theory. I was a student of Professor Tomita from 1966 to 1971, and at that time, he did not mention to us anything about Tomita-Takesaki theory. He would come into the lecture room with a few sticks of chalks and lecture on research topics like “observable algebras”, “operators and operator algebras on Krein spaces” and “noncommutative Fourier analysis” that were elaborated after the Tomita theory. At times, he was standing in front of the blackboard thinking and writing and suddenly he erased everything; it is certain that he was testing his mathematical thoughts of that moment. Personally, I could understand almost nothing of the contents of Tomita’s lectures, so I was just keeping notes. However, I think that I naturally learned from him how to think about mathematics and how to approach mathematical problems. About 10 years later, I noticed that my research topic “unbounded operator algebras” is concerned with the theory of “observable algebras” presented in his aforementioned lectures. Thus, I started attending a lecture again for about 1 year. At that time, I got a copy of the previous lecture notes kept by my colleague H. Kurose, a student of Professor Tomita too, and read it myself in order to use in my research [16]. But, Tomita was not interested in publishing his results. So, as regards his research on “operators and operator algebras on Krein spaces”, Y. Nakagami collaborated with him and the paper “Triangular Matrix Representation for Self-Adjoint Operators in Krein Spaces” was published in Japanese J. Math. in 1988 [23]. Nakagami continued his studies as the only author, resulting in the papers [21, 22]. Another of Tomita’s students, S. Ôta, studied “Lorentz algebras on Krein spaces” [24]. The theory of observable algebras is closely related to operator algebras and its related fields. The Tomita-Takesaki theory is a special case of this theory, every observable algebra can vii

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Preface

be regarded as an operator algebra on a Pontryagin space with codimension 1 and the representation theory of locally convex ∗-algebras results in this theory. From all these, one concludes that this theory provides the mathematical techniques that establish intimate connections between the operator algebras and quantum theories. Unfortunately, Professor Tomita passed away in 2015 without publishing his work on observable algebras. Afraid that this theory would perish without being used, I decided to write these notes based on his research materials: 1. 2. 3. 4.

Notes of Professor Kurose and myself from Tomita’s lectures. Harmonic analysis on topological ∗-algebras [40]. Algebra of observables in Hilbert space [41]. Fundamental of noncommutative Fourier analysis [42].

There were many unproved parts, as well unclear parts in the preceding sources. This note is a compilation of the Tomita’s observable algebras within the scope of what the author, who has been studying them for many years, could achieve. First, I would like to thank my supervisor Professor M. Tomita for giving me many mathematical ideas and for his constant warm generous attention. Many thanks are also due to Professor Y. Nakagami for checking this manuscript in detail and for giving me much advice and comments. Furthermore, I would like to thank Professors M. Fragoulopoulou, S. Ôta and M. Uchiyama for many useful and helpful suggestions. I was able to complete this manuscript by giving lectures to Dr. H. Inoue and Dr. M. Takakura on the contents of the present work, in Fukuoka University, as well as having many discussions with them. I would like to thank H. Inoue for his careful reading of my manuscript and comments. Fukuoka, Japan July 2020

Atsushi Inoue

Contents

1

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

1

2 Fundamentals of Observable Algebras . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Q∗ -algebras and T ∗ -algebras . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Structure of CQ∗ -algebras .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Functional Calculus for Self-adjoint Trio Observables .. . . . . . . . . . . . . . 2.4 ∗-Automorphisms of Observable Algebras . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Locally Convex Topologies on T ∗ -algebras . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6 Commutants and Bicommutants of T ∗ -algebras ... . . . . . . . . . . . . . . . . . . .

5 5 13 16 32 39 41

3 Density Theorems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Von Neumann Type Density Theorem . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Kaplansky Type Density Theorem.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

53 53 62

4 Structure of CT ∗ -Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Decomposition of CT ∗ -Algebras .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Classification of CT ∗ -Algebras . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Commutative Semisimple CT ∗ -Algebras .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Some Results Obtained from T ∗ -Algebras.. . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.1 Projections Defined by T ∗ -Algebras .. . . . .. . . . . . . . . . . . . . . . . . . . 4.4.2 The Vector Representation of the CT ∗ -Algebra Generated by a T ∗ -Algebra .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.3 Construction of Semisimple CT ∗ -Algebras from a T ∗ -Algebra . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.4 Semisimplicity and Singularity of T ∗ -Algebras .. . . . . . . . . . . . . 4.4.5 A Natural Weight on the von Neumann Algebra Defined from a T ∗ -Algebra .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.6 Density of a ∗-Subalgebra of a T ∗ -Algebra . . . . . . . . . . . . . . . . . .

69 70 77 93 98 98 98 106 110 112 116

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Contents

5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Standard T ∗ -Algebras and the Tomita-Takesaki Theory . . . . . . . . . . . . . 5.2 Admissible Invariant Positive Invariant Sesquilinear Forms on a ∗-Algebra .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.1 T ∗ -Algebras Generated by Admissible i.p.s. Forms . . . . . . . . . 5.2.2 Admissibility and Representability of i.p.s. Forms .. . . . . . . . . . 5.2.3 Strongly Regular i.p.s. Forms.. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.4 Decomposition of i.p.s. Forms.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.5 Regularity and Singularity of i.p.s. Forms . . . . . . . . . . . . . . . . . . . . 5.3 Positive Definite Generalized Functions in Lie Groups . . . . . . . . . . . . . . 5.4 Weights in C ∗ -Algebras .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

125 125 131 132 134 141 146 147 151 158

A Functional Calculus, Polar Decomposition and Spectral Resolution for Bounded Operators on a Hilbert Space .. . . . . . . . . . . . . . . . . A.1 Spectrum .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.2 Continuous Functional Calculus of a Self-Adjoint Bounded Linear Operator .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.3 Polar Decomposition for a Bounded Linear Operator . . . . . . . . . . . . . . . . A.4 Spectral Resolution for a Bounded Self-Adjoint Operator . . . . . . . . . . . A.5 Functional Calculus with Borel Functions . . . . . . . .. . . . . . . . . . . . . . . . . . . .

165 166 167 168

B Spectral Resolution of an Unbounded Self-Adjoint Operator and Polor Decomposition of a Closed Operator in a Hilbert Space . . . . B.1 Basic Definitions and Results for Unbounded Linear Operators .. . . . B.2 Spectral Resolution of Unbounded Self-Adjoint Operators . . . . . . . . . . B.3 Functional Calculus for Unbounded Self-Adjoint Operators.. . . . . . . . B.4 Polar Decomposition of Closed Linear Operators . . . . . . . . . . . . . . . . . . . .

171 171 174 174 175

C Banach ∗-Algebras, C ∗ -Algebras, von Neumann Algebras and Locally Convex ∗-Algebras . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . C.1 Banach ∗-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . C.2 C ∗ -Algebras and von Neumann Algebras .. . . . . . . .. . . . . . . . . . . . . . . . . . . . C.3 Density Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . C.4 Locally Convex ∗-Algebras .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

177 177 178 179 180

165 165

References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 181 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 183

Chapter 1

Introduction

This note is a compilation of Tomita’s observable algebras which the author has been studying for many years, based on the Tomita’s lectures at Kyushu University and his RIM-workshop records [40, 41]. The theory of observable algebras is one of the mathematical techniques that establish intimate connections between the theory of operator algebras and quantum theories. In quantum mechanics, self-adjoint operators A in a Hilbert space H with inner product (·|·) represent observables describing a given quantum system, while unit vectors x in H represent states of the system. The value (Ax|x) for any state x is the expectation of the observable A on the state x. In the algebraic approach to quantum mechanics a ∗-algebra on H such as C ∗ -algebra or von Neumann algebra is often assigned to a given quantum mechanics system, and it has been studied from the mathematical aspects as well from the applications to quantum physics. The case that the correspondence of an operator observable A in a quantum system to a state x is a mapping has been considered often. If nothing like that happens in a physical phenomenon, how then can one approach this phenomenon mathematically? Tomita’s idea for this question is to consider the quartet (A, x, y ∗ , μ) consisting of an observable A, of two states x, y and of an expectation μ as an observable, which is called a quartet observable on H, where y ∗ is an element of the dual H∗ of H. This means that an operator observable A having two different states can be regarded as two different observables. From this approach of quartet observables, various types of observables are introduced and considered. The second idea is to introduce an algebraic and a topological structure into a set of quartet observables as follows: let H be a Hilbert space, H∗ the dual of H and B(H) the C ∗ -algebra of all bounded linear operators on H. Let Q∗ (H) be the set of all quartet observables on H. Define in Q∗ (H) the following operations A + B, αA, AB, the involution A → A and the norm A as follows: for A = (A0 , x, y ∗ , α), B = (B0 , u, v ∗ , β) ∈ Q∗ (H) and λ ∈ C A + B = (A0 + B0 , x + u, y ∗ + v ∗ , α + β), λA = (λA0 , λx, λy ∗ , λα), © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Inoue, Tomita’s Lectures on Observable Algebras in Hilbert Space, Lecture Notes in Mathematics 2285, https://doi.org/10.1007/978-3-030-68893-6_1

1

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1 Introduction

AB = (A0 B0 , Ao u, (B0∗ y)∗ , (u|v)), A = (A∗0 , y, x ∗ , α), ¯ A = max(A0 , x, y, |α|). As seen above, a part of the dual state in a quartet observable is necessary for defining the algebraic structure (multiplication, involution) on Q∗ (H). From now on, we denote a quartet observable A = (A0 , x, y ∗ , α) by A = (π(A), λ(A), λ∗ (A), μ(A)). Then Q∗ (H) is a Banach ∗-algebra without identity, π is a bounded ∗-homomorphism of the Banach ∗-algebra Q∗ (H) onto the C ∗ algebra B(H), λ is a bounded linear mapping of Q∗ (H) onto H satisfying λ(AB) = π(A)λ(B) for all A, B ∈ Q∗ (H), λ∗ is a bounded linear mapping of Q∗ (H) onto H∗ satisfying λ∗ (A) = λ(A )∗ and λ∗ (AB) = (π(B)∗ λ(A ))∗ for all A, B ∈ Q∗ (H) and μ is a bounded linear functional on Q∗ (H) satisfying μ(B  A) = (λ(A)|λ(B)) for all A, B ∈ Q∗ (H). A ∗-subalgebra of Q∗ (H) is called a Q∗ -algebra on H and a closed ∗-subalgebra of Q∗ (H) is called a CQ∗ algebra on H. Tomita had explained the physical reason why he introduced such quartet observables. The start was a certain question to the Heisenberg and the von Neumann works about classical quantum mechanics. An observable means an observable part of a physical phenomenon. Heisenberg considered that an observable is expressed as a sequence (aij ) of scalars. We call this a sequential observation. It is quite subjective since different observations must be possible. von Neumann considered that Heisenberg’s sequential observation (aij ) must be a representation of an operator A0 on a Hilbert space H: (aij ) = ((A0 ej |ei )) for some ONB {ei } in H. Besides sequential observations, function observations are frequently used representing an observable to a certain space. He developed operator analysis in Hilbert space considering that an operator is an observable and an operator algebra is a dynamical system. But, it seems that the fundamental type of observables is an observable representation in our sense. For instance, the sequential observation (aij ) of a Hilbert-Schmidt operator A0 on H is different by taking the method of ONBs in H. Because of this fact we consider the quartet observables. Indeed, we denote by H ⊗ H¯ the Hilbert space of all Hilbert-Schmidt operators on H with inner product defined by the trace of S ∗ T . For A0 ∈ H ⊗ H¯ we consider the ¯ quartet observable A = (π(A), λ(A), λ(A )∗ , μ(A)) on the Hilbert space H ⊗ H,  ) = A∗ , where π(A) = (a ) (the Hilbert-Schmidt matrix of A ), λ(A) = A , λ(A ij 0 0 0  μ(A) = ∞ i=1 aii . Then the map of the sequential observation (aij ) to the quartet observable A is injective. We give a physical example of observables and observable algebras in Example 2.1.4. Tomita also introduced other observables and observable algebras as follows: for any A0 ∈ B(H) and x, y ∈ H, the trio (A0 , x, y ∗ ) is called a trio observable, and the set T ∗ (H) of all trio observables on H is a Banach ∗-algebra without identity equipped with the following operations A + B, αA, AB,

1 Introduction

3

the involution A → A and the norm A defined as follows: for A = (A0 , x, y ∗ ), B = (B0 , u, v ∗ ) and α ∈ C A + B = (A0 + B0 , x + u, y ∗ + v ∗ ), λA = (λA0 , λx, λy ∗ ), AB = (A0 B0 , Ao u, (B0∗ y)∗ ), A = (A∗0 , y, x ∗ ), A = max(A0 , x, y). A trio observable A = (A0 , x, y ∗ ) is denoted by A = (π(A), λ(A), λ∗ (A)). Then π, λ and λ∗ have the same properties as those in the case of Q∗ (H). A ∗-subalgebra of the Banach ∗-algebra T ∗ (H) is called a T ∗ -algebra on H and a closed ∗subalgebra of T ∗ (H) is called a CT ∗ -algebra on H. In Chap. 2 we show that every CQ∗ -algebra is divided into two types by its expectation, and see that the study of CQ∗ -algebras results in that of CT ∗ -algebras. So we mainly deal with T ∗ -algebras and CT ∗ -algebras. The main theme of this note is to investigate the relationship between the operator part π(A) and the vector part λ(A) of a CT ∗ -algebra A on H. The correspondence π(A) → λ(A) is not necessarily even a mapping. Using the fundamental theory of CT ∗ -algebras in Chaps. 2 and 3, we can prove that (i) the correspondence π(A) → λ(A) is a uniformly continuous mapping from the ∗-algebra π(A) on H to the Hilbert space H if and only if there exists a vector g in H such that λ(A) = π(A)g for all A ∈ A; (ii) the correspondence π(A) → λ(A) is a uniformly closable  mapping if and only if there exists a subset {gα } of H such that λ(A) = α π(A)gα for all A ∈ A, where {gα } is a subset of H satisfying (π(A)gα |gβ ) = 0 for α = β and α π(A)gα 2  λ(A)2 for all A ∈ A; (iii) λ(N(A)) is dense in H if and only if for any x ∈ H there exists a sequence {An } in A such that π(An ) = 0 for all n ∈ N and limn→∞ λ(An ) = x, where N(A) := {A ∈ A; π(A) = 0}; (iv) every CT ∗ -algebra is decomposed into a CT ∗ -algebra satisfying (ii) and a CT ∗ -algebra satisfying (iii). In a physical sense, we regard a CT ∗ -algebra A as a quantum mechanics system, its operator part π(A) as an operator observable algebra, its vector part λ(A) as a state space and its dual vector part λ∗ (A) as a dual state space. It can be explained that a physical phenomenon is regular in case of (ii), and nonregular in case of (iii), and every physical phenomenon can be decomposed into the regular part and the nonregular part. Chapters 2 and 3 build the foundations of the theory of T ∗ -algebras and CT ∗ algebras. More precisely, Chap. 2 deals with a spectrum of a trio observable, a functional calculus of (self-adjoint) trio observable A, the square root of A A and the polar decomposition of A. After defining several locally convex topologies called weak, σ -weak, strong, strong∗ and σ -strong∗ on T ∗ (H), we introduce three

4

1 Introduction

commutants Aπ , Aτ and Aρ and the bicommutants Aππ , Aτ τ and Aρρ of a T ∗ algebra A. In Chap. 3 we treat the von Neumann type density theorem and the Kaplansky type density theorem. Both play an important role in the study of observable algebras. Chapter 4 presents one of the major achievements of the theory, namely, the investigation of the relationship between the operator representation and the vector representation of a CT ∗ -algebra. Let A be a CT ∗ -algebra on H. We define the closed ∗-ideals P (A) and N(A) of A by the closed linear span of the sets {AB; A, B ∈ A} and {A ∈ A; π(A) = 0}, respectively. Let PA and NA be the projections onto the closed subspace of H generated by the set π(A)H and onto the closed subspace of H generated by the set λ(N(A)), respectively. Then PA ∈ π(A) ∩ π(A) and NA ∈ π(A) . Let FA = PA (I − NA ), and let F (A) = {(π(A), FA λ(A), (FA λ(A ))∗ ); A ∈ A}. Then we see that F (A) is a CT ∗ -algebra on H. If FA = I , A = F (A) and A = N(A), then A is said to be regular, semisimple and singular, respectively, and it is proved that A is decomposed into the semisimple CT ∗ -algebra F (A) and the singular CT ∗ algebra N(A) as follows: A = F (A) + N(A), which shows that the aforementioned statement (iv) holds. Furthermore, proving the equalities: PA = I − NAτ and PAτ = I − NA , we get one of the main results in this note: A is semisimple if and only if the aforementioned statement (ii) holds, and A is singular if and only if the aforementioned statement (iii) holds. This theory of observable algebras is concerned in that of operator algebras and its applications: in Chap. 2 we have seen that every observable algebra can be regarded as an operator algebra on a Pontryagin space of codimension 1. In Chap. 5 it is shown that the Tomita-Takesaki theory is a special case of this theory, the representation theory of (locally convex) ∗-algebras results in this theory, and this theory is applicable to the study of weights on C ∗ -algebras and of positive definite generalized functions in Lie groups. Since the author wrote this note on few research materials containing many unclear parts, there were among them some parts that he could not understand. So this note is a part of the work of Tomita within the range of author’s understanding: therefore there are still many interesting subjects for investigation and thus there is room for further development. This note deals with an observable algebra whose operator representation is bounded, but the operator observables that appear in quantum mechanics are unbounded such as the position and momentum operators, and in addition the operator representation defined by the GNS-construction of a positive linear functional on a ∗-algebra is unbounded in general, From these, it is necessary to study an unbounded observable algebra, namely an observable algebra whose operator representation is unbounded. The ideas of Tomita have already influenced some of author’s work for unbounded operator algebras such as the unbounded Tomita-Takesaki theory [3, 17] and the representation theory of (locally convex) ∗-algebras [3, 5, 16]. However, these studies are not contained in this note except for the author’s study of admissible positive linear functionals on (locally convex) ∗-algebras contained in Sect. 5.1. The author’s motivation was to write this note with the aim that Tomita’s theory be widely known and freely used by researchers in many (unbounded) operator algebras and related fields without being buried. The systematic studies of unbounded observable algebras will be an issue for the future.

Chapter 2

Fundamentals of Observable Algebras

This chapter is devoted to the fundamental theory of observable algebras. In Sect. 2.1 the notions of observables and observable algebras are defined. Section 2.2 deals with structure of CQ∗ -algebras. In Sect. 2.3 we discuss the spectrum of a trio observable and a functional calculus of a (self-adjoint) trio observable, and using it we define the square root and the polar decomposition of a trio observable. Section 2.4 discuss ∗-automorphisms of observable algebras. In Sect. 2.5 we define some locally convex topologies on T ∗ (H) called weak, σ -weak, strong, σ -strong, strong∗ and σ -strong∗ and investigate their properties. In Sect. 2.6 we define and examine three commutants Aπ , Aτ and Aρ , and bicommutants Aππ , Aτ τ and Aρρ of a T ∗ -algebra A.

2.1 Q∗ -algebras and T ∗ -algebras Let H be a Hilbert space with inner product (·|·) and H ∗ the dual of H. For any x ∈ H we define a linear functional x ∗ on H by < x ∗ , y >= (y|x), y ∈ H. Then x ∗ ∈ H ∗ , and by the Riesz theorem the map: x → x ∗ is a conjugate linear isometry of H onto H ∗ . Let B(H) be the C ∗ -algebra of all bounded linear operators on H. For A0 ∈ B(H) and x ∈ H, we define elements x ∗ A0 and A0 x ∗ of H ∗ by x ∗ A0 = (A∗0 x)∗ and A0 x ∗ = (A0 x)∗ . A quartet (A0 , x, y ∗ , r) of A0 ∈ B(H), x, y ∈ H and r ∈ C is called a quartet observable on H.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Inoue, Tomita’s Lectures on Observable Algebras in Hilbert Space, Lecture Notes in Mathematics 2285, https://doi.org/10.1007/978-3-030-68893-6_2

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6

2 Fundamentals of Observable Algebras

Let Q∗ (H) be the set of all quartet observables on H; this is a Banach ∗-algebra without identity equipped with the following operations, involution  and norm: A + B = (A0 + B0 , x + u, y ∗ + v ∗ , r + δ), αA = (αA0 , αx, αy ∗ , αr), AB = (A0 B0 , A0 u, (B0∗ y)∗ , (u|y)), A = (A∗0 , y, x ∗ , r¯ ), A = max(A0 , x, y, |r|) for A = (A0 , x, y ∗ , r), B = (B0 , u, v ∗ , δ) ∈ Q∗ (H) and α ∈ C. For A = (A0 , x, y ∗ , r) ∈ Q∗ (H), write π(A) = A0 , λ(A) = x, λ∗ (A) = y ∗ , μ(A) = r. Then π is a bounded ∗-homomorphism of the Banach ∗-algebra Q∗ (H) onto the C ∗ -algebra B(H) satisfying π(A) ≤ A for all A ∈ Q∗ (H), λ is a bounded linear map of Q∗ (H) onto H satisfying λ(AB) = π(A)λ(B) for all A, B ∈ Q∗ (H), λ∗ is a bounded linear map of Q∗ (H) onto H ∗ satisfying λ∗ (A) = λ(A )∗ , λ∗ (AB) = (π(B)∗ λ(A ))∗ for all A, B ∈ Q∗ (H), and μ is a bounded linear functional on Q∗ (H) satisfying μ(B  A) = (λ(A)|λ(B)) for all A, B ∈ Q∗ (H). We call π, λ, λ∗ and μ the operator representation, the vector representation into H, the vector representation into H ∗ and the expectation of Q∗ (H), respectively. Definition 2.1.1 A ∗-subalgebra of the Banach ∗-algebra Q∗ (H) is called a quartet observable algebra on H, for simplicity, a Q∗ -algebra on H. A closed ∗-subalgebra of the Banach ∗-algebra Q∗ (H) is called a complete quartet observable algebra on H, for simplicity, a CQ∗ -algebra on H. We next define the notions of trio observable algebras. A triple (A0 , x, y ∗ ) of A0 ∈ B(H), x ∈ H and y ∗ ∈ H ∗ is called a trio observable on H, and denote

2.1 Q∗ -algebras and T ∗ -algebras

7

by T ∗ (H) the set of all trio observables on H. Then T ∗ (H) is a Banach ∗-algebra without identity equipped with the following operations, involution and norm: A + B = (A0 + B0 , x + u, y ∗ + v ∗ ), αA = (αA0 , αx, αy ∗ ), AB = (A0 B0 , A0 u, (B0∗ y)∗ ), A = (A∗0 , y, x ∗ ), A = max(A0 , x, y) for A = (A0 , x, y ∗ ), B = (B0 , u, v ∗ ) and α ∈ C. For A = (A0 , x, y ∗ ) ∈ T ∗ (H), write π(A) = A0 ,

λ(A) = x and λ∗ (A) = y ∗ .

Then π is a bounded ∗-homomorphism of the Banach ∗-algebra T ∗ (H) onto B(H), λ is a bounded linear map of T ∗ (H) onto H satisfying λ(AB) = π(A)λ(B) for all A, B ∈ T ∗ (H), and λ∗ is a bounded linear map of T ∗ (H) onto H ∗ satisfying λ∗ (A) = λ(A )∗ , λ∗ (AB) = (π(B)∗ λ(A ))∗ for all A, B ∈ T ∗ (H). Definition 2.1.2 A ∗-subalgebra of the Banach ∗-algebra T ∗ (H) is called a trio observable algebra on H, for simplicity, a T ∗ -algebra on H. A closed ∗-subalgebra of the Banach ∗-algebra T ∗ (H) is called a complete trio observable algebra on H, for simplicity, a CT ∗ -algebra on H. Let A be a T ∗ -algebra (or a Q∗ -algebra) on H. The norm topology on the Banach ∗-algebra A is called the uniform topology on A and is denoted by τu . Let A and B be T ∗ -algebras (or Q∗ -algebras) on H. If there exists an algebraic ∗-isomorphism and isometry of A[τu ] onto B[τu ], then we say that the T ∗ -algebras (or Q∗ -algebras) A and B are isomorphic. We investigate the relationship between Q∗ (H) and T ∗ (H). An element V := (0, 0, 0, 1) of Q∗ (H) is called the vacuum (or, unit expectation) of Q∗ (H), and CV := {αV ; α ∈ C} is identical with the radical of Q∗ (H), that is, CV = {A ∈ Q∗ (H); AX = XA for all X ∈ Q∗ (H)}.

8

2 Fundamentals of Observable Algebras

Hence CV is a closed ∗-ideal of Q∗ (H). Put τ (A) = (π(A), λ(A), λ∗ (A)), A ∈ Q∗ (H). Then τ is a bounded ∗-homomorphism of Q∗ (H) onto T ∗ (H) whose kernel is CV . Hence T ∗ (H) is isomorphic to the quotient Banach ∗-algebra Q∗ (H)/CV as Banach ∗-algebras. Here we give two concrete examples of observable algebras. The first is of observable algebras whose elements are certain functions on R. Example 2.1.3 We denote by C(R), Cb (R), L1 (R) and L2 (R) the space of all continuous, bounded continuous, integrable and square integrable functions on R, respectively. (1) We put At = {ft = (π(f ), λ(f ), λ(f¯)∗ ); f ∈ Cb (R) ∩ L2 (R)}, Aq = {fq = (π(f ), λ(f ), λ(f¯)∗ , μ(f )); f ∈ Cb (R) ∩ L1 (R)}, where π(f )g = fg, g ∈ L2 (R), λ(f ) = f , λ∗ (f ) = f¯∗ (f¯(x) = f (x)) and μ(f ) = R f (x)dx. Then At is a CT ∗ -algebra on L2 (R) which is isomorphic to the Banach ∗-algebra Cb (R) ∩ L2 (R) with the usual function operations, involution and the norm f u∧2 := max(f u , f 2 ), where f u = maxt ∈R |f (t)|. Aq is a CQ∗ -algebra on L2 (R) which is isomorphic to the Banach ∗-algebra Cb (R) ∩ L1 (R) with the usual function operations, involution and the norm f u∧1 := max(f u , f 1 ). (2) We put Bt = {ft = (π(f ), λ(f ), λ(f¯)∗ ); f ∈ L1 (R) ∩ L2 (R)}, Bq = {fq = (π(f ), λ(f ), λ(f¯)∗ , f (0)); f ∈ L1 (R) ∩ L2 (R) ∩ C(R)},  where π(f )q = f ∗q (the convolution operator, (f ∗g)(x) = f (t)g(t −x)dt), ∗ λ(f ) = f , λ∗ (f ) = f (−x) and μ(f ) = f (0). Then Bt is a CT ∗ -algebra on L2 (R) which is isomorphic to the Banach ∗-algebra L1 (R) ∩ L2 (R) with the convolution multiplication f ∗g, the involution f¯(−x) and the norm f 1∧2 := max(f 1 , f 2 ). Bq is a CQ∗ -algebra on L2 (R) which is isomorphic to the Banach ∗-algebra with same multiplication and involution as those of Bt and the norm f u∧1∧2 := max(f u , f 1 , f 2 ). The second is of observable algebras related to the CCR-algebra of degree 1. Example 2.1.4 Let A be a ∗-algebra generated by identity 11 and two Hermitian elements p and q satisfying the Heisenberg commutation relation: [p, q] := pq − qp = −i11

(2.1.1)

2.1 Q∗ -algebras and T ∗ -algebras

9

and it is called a CCR-algebra of degree 1. It is not possible to find Hermitian matrices P and Q satisfying the Heisenberg commutation, however Heisenberg found a solution in the form of infinite matrices: ⎞ ⎛ 0 1 √0 0 0 ··· ⎜ −1 0 2 √0 0 ···⎟ ⎟ √ −i ⎜ ⎟ ⎜ P = √ ⎜ 0 − 2 0 3 0 · · · ⎟, √ √ ⎟ 2⎜ ⎝ 0 0 4 ···⎠ 0 − 3 √ 0 0 0 − 4 0 ··· ⎞ ⎛ 0 1 √0 0 · · · 1 ⎜1 0 2 √0 · · · ⎟ ⎟. √ (2.1.2) Q= √ ⎜ ⎝ 3 ···⎠ 2 0 2 √0 3 0 ··· 0 0 Schrödinger found another solution to the Heisenberg solution in his formulation of quantum mechanics. The operators PS and QS defined by d f (t), dt (QS f )(t) = tf (t) (PS f )(t) = −i

for every function f in the Schwartz space S(R) are essentially self-adjoint operators in L2 (R) satisfying the Heisenberg commutation relation. We here denote by the same PS and QS the closures of PS and QS , which are called the momentum and position operators, respectively. He showed that PS and QS have the same matrices as in Eq. (2.1.2) with respect to the ONB for L2 (R) defined by ϕn (t) = π − 4 (2n n!)− 2 (t − 1

1

d n − t2 ) e 2. dt

This representation is called the Schrödinger representation of the CCR-algebra A . This raised the question as to whether there were other ways to realize the Heisenberg commutation relations. Weyl proposed the question should be formulated in terms of strongly continuous unitary groups U (s) = eisP and V (t) = eit Q satisfying the commutation relation: U (s)V (t) = eist V (t)U (s)

(2.1.3)

for all s, t ∈ R. Von Neumann solved the question posed by Weyl. He showed that (US (t)f )(x) := (eit PS f )(x) = f (x + t), (VS (t)f )(x) := (eit QS f )(x) = eit x f (x)

10

2 Fundamentals of Observable Algebras

for all f ∈ L2 (R) and t ∈ R and {US (s), VS (t); s, t ∈ R} satisfies the Weyl commutation relation (2.1.3), and every strong continuous unitary groups {U (s); s ∈ R} and {V (t); t ∈ R} on H satisfying the Weyl commutation relation is unitarily equivalent to a direct sum of a sequence of the unitary groups {US (s); s ∈ R} and {VS (t); t ∈ R}. The unitary groups {US (s); s ∈ R} and {VS (t); t ∈ R} defined by the momentum operator PS and the position operator QS are called motion groups. Here we remark that the unitary groups {eisP , eit Q ; s, t ∈ R} defined by self-adjoint operators P and Q satisfying the Heisenberg commutation relation don’t necessarily satisfy the Weyl commutation relation. The momentum operator PS and the position operator QS themselves must be observables , as well various observables whose operator parts are defined by PS and QS . Thus there are observables whose operator parts are unbounded. From this, we here after need to consider such observables and observable algebras. This note deals with only observables whose operator parts are bounded and observable algebras consisting of such observables. Hence we consider observable algebras whose operator parts are contained in a ∗-algebra defined by the unitary groups {US (s); s ∈ R} and {VS (t); t ∈ R}. Then we can introduce various types of observables and observable algebras. For instance, let f0 ∈ L2 (R). Then ft := US (t)fo , t ∈ R must be a motive observable, expressing the time process of vectors. This processing must d be the solution of motion equation dt x(t) = iPS x(t). Then we can think that the momentum operator PS is an operator observable, which determines the process of ft by knowing the initial value f0 . We can discuss the same thing for the position operator QS . Let A0 be a ∗-algebra on L2 (R) defined by {US (s); s ∈ R} and {VS (t); t ∈ R}. For any non-zero element f in L2 (R) we put Af := {A = (A0 , A0 f, (A∗0 f )∗ , (A0 f |f )); A0 ∈ A0 }. Then it is a Q∗ -algebra whose operator representation, vector and dual representations and expectation are well-behaved. On the other hand, we put B := {(0, f, g ∗ , α); f, g ∈ L2 (R), α ∈ C}. Then it is a Q∗ -algebra on L2 (R) whose operator, vector, dual vector representations and expectation don’t have any relationships. Finally we show that Q∗ (H) can be regarded as an operator algebra on the Krein space C×H ×C. We first briefly recall some definitions and terminology concerning Krein spaces. Suppose that J is an Hermitian unitary operator on a Hilbert space H with inner product (·|·). Then the Hermitian sesquilinear form < ·, · >J on H × H defined by < x, y >J = (J x|y), x, y ∈ H, is called an indefinite inner product on H defined by J , and H is said to be a Krein space with the indefinite inner product < ·, · >J , and is denoted by HJ . By the

2.1 Q∗ -algebras and T ∗ -algebras

11

Riesz theorem, for any A ∈ B(H) there exists a unique element AJ of B(H) such that < Ax, y >J =< x, AJ y >J for all x, y ∈ H, which satisfies the equality: AJ = J A∗ J. The operator AJ is called the J -adjoint of A. By the spectral resolution of J , there exist projections J+ and J− satisfying J = J+ − J− . The closed subspaces J+ H and J− H of H are denoted by H+ and H− , and their dimensions are denoted by dim H+ and dim H− , respectively. If dim H+ and dim H− are both finite, then the Krein space HJ is called the Minkowski space, and if either dim H+ or H− is finite, then HJ is called the Pontrjagin space. We now consider the product space K := C × H × C, and write an element x of K by ⎛

⎞ μ(x) x = ⎝ λ(x) ⎠ , δ(x) where μ(x), δ(x) ∈ C and λ(x) ∈ H. Then K is a Hilbert space under the usual operations and inner product: (x|y) = μ(x)μ(y) + (λ(x)|λ(y)) + δ(x)δ(y) for all x, y ∈ K, and has an Hermitian unitary operator J on K: ⎛

00 J = ⎝0 I 10

⎞ 1 0⎠. 0

Hence, K is a Krein space with indefinite inner product: < x, y >J = (J x|y) = μ(x)δ(y) + (λ(x)|λ(y)) + δ(x)μ(y) for all x, y ∈ K. Since ⎛

⎞ 1/2 0 1/2 J+ = ⎝ 0 I 0 ⎠ 1/2 0 1/2

⎛

⎞ 1/2 0 −1/2 and J− = ⎝ 0 0 0 ⎠ , −1/2 0 1/2

12

2 Fundamentals of Observable Algebras

it follows that dim K+ = ∞ and dim K− = 1. Thus the Krein space KJ is a Pontrjagin space with codimension 1. An element A of Q∗ (H) is represented as an operator on the Pontrjagin space KJ : ⎛

⎞ 0 λ∗ (A) μ(A) A = ⎝ 0 π(A) λ(A) ⎠ . 0 0 0 The involution A of A ∈ Q∗ (H) coincides with the J -adjoint AJ : ⎛

⎞ 0 λ(A)∗ μ(A) AJ = ⎝ 0 π(A)∗ λ∗ (A)∗ ⎠ . 0 0 0 Thus Q∗ -algebras can be regarded as operator algebras on Pontrjagin spaces with codimension 1. T ∗ -algebras are also related to operator algebras on Pontrjagin spaces. Indeed, ⎛

0 ⎜0 J =⎜ ⎝0 1

0 0 I 0

0 I 0 0

⎞ 1 0⎟ ⎟ 0⎠ 0

is an Hermitian unitary operator on the Hilbert space K = C × H × H × C, and so we can define a Pontrjagin space KJ with indefinite inner product defined by J : < x, y >J = (J x|y) = α4 β¯1 + (x3 |y2 ) + (x2 |y3 ) + α1 β¯4 . Then, an element A of T ∗ (H) is regarded as an operator on the Pontrjagin space KJ : ⎛

⎞ 0 λ∗ (A) 0 0 ⎜ 0 π(A) 0 0 ⎟ ⎜ ⎟ ⎝ 0 0 π(A) λ(A) ⎠ . 0 0 0 0 Notes For Banach ∗-algebras, C ∗ -algebras and the algebraic notions (∗homomorphisms, ∗-isomorphisms, etc.), refer to Appendix C. As for Krein spaces, refer to [23].

2.2 Structure of CQ∗ -algebras

13

2.2 Structure of CQ∗ -algebras In this section we show that a CQ∗ -algebra is divided into two types by its expectation, and that the study of CQ∗ -algebras results in that of CT ∗ -algebras. Lemma 2.2.1 Let A be a Q∗ -algebra (or, a T ∗ -algebra) on a Hilbert space H. If the map: π(A) → λ(A) is uniformly continuous, then there exists an element g of H such that λ(A) = π(A)g for all A ∈ A. Proof Since π(A) → λ(A) is uniformly continuous, there exists a constant γ > 0 such that λ(A)  γ π(A)

(2.2.1)

for all A ∈ A. For any ε > 0 and A1 , A2 , . . . , An ∈ A, we put (ε; A1 , . . . , An ) = {x ∈ H; x  γ , π(Ak )x − λ(Ak )  ε for all k = 1, . . . , n}. Since the uniform closure of π(A) in B(H) is a C ∗ -algebra on H, it has an approximate identity, so that there exists an element A of A such that π(A)  1, π(Ak )π(A) − π(Ak )
0 such that |μ(A)|  γ max(π(A), λ(A), λ(A )) for all A ∈ A. Hence we have λ(A)2 = μ(A A)  γ max(π(A A), λ(A A))  γ max(π(A)2 , π(A)λ(A)), which implies that λ(A)  max(γ , 1)π(A)

(2.2.3)

2.2 Structure of CQ∗ -algebras

15

for all A ∈ A. By Lemma 2.2.1 there exists an element g of H such that λ(A) = π(A)g for all A ∈ A, which yields by (2.2.3) that |μ(A)|  (γ + g)π(A)

(2.2.4)

for all A ∈ A. Take an arbitrary A ∈ A. As shown in (2.2.2), for any ε > 0 we can find an element B of A such that π(B)π(A) − π(A) < ε Hence it follows from (2.2.4) that |(π(A)g|g) − μ(A)|  |(π(A)g|g) − (π(B)π(A)g|g)| +|(π(B)π(A)g|g) − μ(A)| = |((π(B)π(A) − π(A))g|g)| +|μ(BA) − μ(A)|  π(B)π(A) − π(A)g2 + max(γ , g)π(B)π(A) − π(A) < (g2 + max(γ , g))ε, so that μ(A) = (π(A)g|g), which proves (iii). (iii)⇒(iv) This follows from π(A)  A  max(1, g2 )π(A) for all A ∈ A. Thus, (i)–(iv) are equivalent. Suppose that A is a CQ∗ -algebra on H. (iv)⇒(v) This is trivial. (v)⇒(i) Putting μ(π(A)) ˜ = μ(A), A ∈ A, μ˜ is a positive linear functional on the C ∗ -algebra π(A). Hence it follows from [37, Proposition 9.12] that μ˜ is continuous, which derives (i). This completes the proof. Lemma 2.2.3 Let A be a CQ∗ -algebra on H and μ its expectation. If the linear functional on the normed ∗-algebra τ (A) defined by τ (A) → μ(A) is not continuous, then CV is a closed ∗-ideal of A, and τ (A) is a CT ∗ -algebra on H which is isomorphic to A/CV .

16

2 Fundamentals of Observable Algebras

Proof Since τ (A) → μ(A) is not continuous, we can choose a sequence {Bn } in A such that limn→∞ τ (Bn ) = 0 and μ(Bn )  1, n = 1, 2, . . .. We here put An =

Bn , μ(Bn )

n = 1, 2, . . . .

Then {An } ⊂ A, limn→∞ τ (An ) = 0 and μ(An ) = 1, n = 1, 2, . . .. Therefore, lim An = lim (τ (An ), μ(An )) = V .

n→∞

n→∞

Since A is a CQ∗ -algebra, it follows that V ∈ A and CV is a closed ∗-ideal of A. Hence, τ (A) := {(π(A), λ(A), λ∗ (A)); A ∈ A} is isomorphic to A/CV as a Banach ∗-algebra. Thus, τ (A) is a CT ∗ -algebra on H. This completes the proof. By Lemmas 2.2.2 and 2.2.3 we have the following Theorem 2.2.4 Let A be a CQ∗ -algebra on H and μ its expectation. Then (1) μπ is finite if and only if there exists an element g of H such that A = (π(A), π(A)g, (π(A)∗ g)∗ , (π(A)g|g)) for all A ∈ A, (2) μπ is infinite if and only if CV is a closed ∗-ideal of A and the CT ∗ -algebra τ (A) is isomorphic to the CT ∗ -algebra A/CV . In case (1) in Theorem 2.2.4 a CQ∗ -algebra A is decided by π(A), and in case (2) it is decided by the CT ∗ -algebra τ (A).

2.3 Functional Calculus for Self-adjoint Trio Observables We begin with a spectrum of trio observable and a functional calculus of a selfadjoint trio observable. For A ∈ T ∗ (H) we denote by CT ∗ (A) the CT ∗ -algebra generated by A. Let T11∗ (H) denote the adjunction of identity 11 to T ∗ (H), and it is also called the unitization of T ∗ (H). We define the spectrum of A ∈ T ∗ (H) as follows: Sp (A) = {λ ∈ C; (A − λ11)−1 does not exist in T11∗ (H)}. Proposition 2.3.1 Let A ∈ T ∗ (H). Then Sp (A) = Sp (π(A)) ∪ {0},

2.3 Functional Calculus for Self-adjoint Trio Observables

17

where Sp (π(A)) is the spectrum of π(A) defined by Sp (π(A)) = {λ ∈ C;

(π(A) − λI )−1 does not exist in B(H)}.

Proof Suppose 0 ∈ Sp (A). Then there exist α ∈ C and B ∈ T ∗ (H) such that (B + α11)A = 11, which implies that 11 = BA + αA ∈ T ∗ (H). This contradicts that T ∗ (H) does not have identity. Hence we have 0 ∈ Sp (A).

(2.3.1)

Next, take an arbitrary non-zero λ ∈ Sp (A). Then there exist α ∈ C and B ∈ T ∗ (H) such that (B − α11)(A − λ11) = (A − λ11)(B − α11) = 11, so that αλ = 1 and BA − λB −

1 1 A = AB − A − λB = O, λ λ

which implies that (π(B) −

1 1 I )(π(A) − λI ) = (π(A) − λI )(π(B) − I ) = I. λ λ

This means that (π(A) − λI )−1 = (π(B) − λ1 I ). Thus, λ ∈ Sp (π(A)), so we get Sp (π(A)) ⊂ Sp (A).

(2.3.2)

Conversely, take an arbitrary non-zero λ ∈ Sp (π(A)). Then there exists an element B0 of B(H) such that (B0 −

1 1 I )(π(A) − λI ) = (π(A) − λI )(B0 − I ) = I. λ λ

We here write 1 1 (B0 λ(A) − λ(A)), λ λ 1 y = ((π(A) − λI )−1 )∗ λ(A )), λ x=

and B = (B0 , x, y ∗ ).

18

2 Fundamentals of Observable Algebras

Then it follows that B ∈ T ∗ (H) and (B −

1 1 11)(A − λ11) = (A − λ11)(B − 11) = 11, λ λ

which yields that λ ∈ Sp (A). Therefore Sp (A) ⊂ Sp (π(A)) ∪ {0}.

(2.3.3)

By (2.3.1)–(2.3.3) we have Sp (A) = Sp (π(A)) ∪ {0}. This completes the proof. We first consider a functional calculus of self-adjoint trio observables. Let A = A ∈ T ∗ (H). Then π(A)∗ = π(A) and λ∗ (A) = λ(A)∗ . By Proposition 2.3.1 and Appendix A.1 Sp (A) is a compact subset of R and is contained in [−π(A), π(A)]. For any polynomial on R: p(t) = α0 + α1 t + · · · + αn t n , we define an element p(A) of T11∗ (H) as follows: p(A) = α0 11 + α1 A + · · · + αn An . Then we have p(A) = α0 11 + (α1 π(A) + · · · + αn π(A)n , (α1 I + α2 π(A) + · · · + αn π(A)n−1 )λ(A), ((α¯ 1 I + α¯ 2 π(A) + · · · + α¯ n π(A)n−1 )λ(A))∗ ) = p(0)11 + (pπ (π(A)), pλ (π(A))λ(A), (p¯ λ (π(A))λ(A))∗ ), where pπ (t) := p(t) − p(0),  pλ (t) := α1 + α2 t + · · · + αn t

n−1

=

pλ (t) := pλ (t).

p(t )−p(0) , t p (0),

t=  0, t = 0,

(2.3.4)

2.3 Functional Calculus for Self-adjoint Trio Observables

19

It is clear that p(A) ∈ T ∗ (H) if and only if p(0) = 0. Based on this fact, we consider a functional calculus for a Banach ∗-algebra Cδ (Sp (A)) of continuous functions on Sp (A) defined below: Let f be a continuous function on Sp (A) vanishing at 0 and having the formal derivative f (0) at 0. Then we have f u := sup |f (t)| = f (π(A)).

(2.3.5)

t ∈Sp (A)

As the same as (2.3.4), we define a Borel function fλ on Sp (A) by  fλ (t) =

f (t ) t , f (0),

t = 0 t = 0.

Then if fλ (π(A))λ(A) is well-defined, then fλ μ := fλ (π(A))λ(A) =

1 |fλ (t)| dμA (t) 2

2

< ∞,

(2.3.6)

 where π(A) = tdE(t) is the spectral resolution of π(A), and μA is the bounded Borel measure on R defined by μA (t) = (E(t)λ(A)|λ(A)) (see Appendix A.4). However, we don’t know whether fλ (π(A))λ(A) is well-defined in general. We now denote by Cδ (Sp (A)) the set of all continuous functions f on Sp (A) with formal derivative f (0) at 0 such that f (0) = 0 and fλ μ < ∞. For any f ∈ Cδ (Sp (A)) we can define an element f (A) of T ∗ (H) as follows:   f (A) = f (π(A)), fλ (π(A))λ(A), (f¯λ (π(A))λ(A))∗ .

(2.3.7)

It is easily shown that Cδ (Sp (A)) is a Banach space under the usual function operations f + g, αf and the norm: f  := max(f u , fλ μ ).

(2.3.8)

(fg)λ = fgλ and (f ∗ )λ = (f¯)λ ,

(2.3.9)

Furthermore, since

where f¯(t) = f (t), t ∈ Sp (A), we have fgu  f u gu  f g, (fg)λ μ = f (π(A))gλ (π(A))λ(A)  f u gλ μ

20

2 Fundamentals of Observable Algebras

 f g, ∗

f u = f u and (fλ )∗ μ = fλ μ , which shows that Cδ (Sp (A)) is a Banach ∗-algebra. Furthermore, it follows from (2.3.5)–(2.3.8) that f (A) = max (f (π(A)), fλ (π(A))λ(A))   = max f u , fλ μ = f 

(2.3.10)

for all f ∈ Cδ (Sp (A)). The following theorem for the functional calculus for Cδ (Sp (A)) is one of the main results in this section. Theorem 2.3.2 Let A = A ∈ T ∗ (H). For any f ∈ Cδ (Sp (A)) we put   f (A) = f (π(A)), fλ (π(A))λ(A), (f¯λ (π(A))λ(A))∗ . Then the map: f → f (A) is a ∗-isomorphism of the Banach ∗-algebra Cδ (Sp (A)) onto the Banach ∗-algebra CT ∗ (A), that is, for any f, g ∈ Cδ (Sp (A)) and α ∈ C we have (f + g)(A) = f (A) + g(A), (αf )(A) = αf (A), (fg)(A) = f (A)g(A), f ∗ (A) = f (A) , f  = f (A). To prove this theorem, we prepare some statements. Let α ∈ Sp (A) and put qα (t) = t (t − α)−1 , t ∈ Sp (A). Then it is clear that qα ∈ Cδ (Sp (A)). Lemma 2.3.3 For any α ∈ Sp (A), qα (A) ∈ CT ∗ (A). Proof that if |α| > π(A), then qα is analytic on Sp (A) and qα (t) =  It ist clear n ∗ − ∞ n=1 ( α ) . Hence, qα (A) ∈ CT (A). Take an arbitrary α ∈ Sp (A) such that ∗ qα (A) ∈ CT (A). Then we have αqα (t) − βqβ (t) = (α − β)qα (t)qβ (t),

t ∈ Sp (A)

(2.3.11)

2.3 Functional Calculus for Self-adjoint Trio Observables

21

for each β ∈ Sp (A), which implies qβ (A) ∈ CT ∗ (A) for some neighborhood of α. Furthermore, since C−Sp (A) is connected, it follows from the analytic continuation that qα (A) ∈ CT ∗ (A) for all α ∈ Sp (A). Let M be the closed subspace of Cδ (Sp (A)) generated by {qα ; α ∈ Sp (A)}. By (2.3.11) and Lemma 2.3.3, we can prove qα qβ =

1 (αqα − βqβ ) ∈ M α−β

for any α, β ∈ Sp (A) with α = β. Hence M is a closed ∗-subalgebra of the Banach ∗-algebra Cδ (Sp (A)). We will verify that M = Cδ (Sp (A)) under some preparations. Lemma 2.3.4 Suppose that f is an analytic function on Sp (A) such that f (0) = 0. Then f ∈ M. Proof Since g(t) :=

f (t ) t

is an analytic function on Sp (A) by f (0) = 0, we have

f (t) = tg(t) =

1 2πi

1 =− 2πi

|α−t |=1

tg(α) dα α−t

|α−t |=1

g(α)qα (t)dα

∈ M. Lemma 2.3.5 For any f ∈ Cδ (Sp (A)) and α ∈ Sp (A) we have f qα ∈ M. Proof For any polynomial p(t) = α0 + α1 t + · · · + αn t n , we can prove pqα ∈ M. Indeed, it follows from Lemma 2.3.4 that r(t) := α1 t + · · · + αn t n ∈ M, which implies since M is an algebra that pqα = α0 qα + rqα ∈ M. Take an arbitrary f ∈ Cδ (Sp (A)). By the Weierstrass approximation theorem, there exists a sequence {pn } of polynomials which converges uniformly to f on Sp (A). Then, since {pn qα } converges uniformly to f qα on Sp (A) and pn qα , f qα ∈ Cδ (Sp (A)), n = 1, 2, . . ., we have lim pn qα − f qα u = 0.

n→∞

Furthermore, the equalities: (pn qα )λ = (pn )λ qα + pn (0)(qα )λ and (f qα )λ = fλ qα ,

22

2 Fundamentals of Observable Algebras

implies that (pn qα )λ − (f qα )λ μ =  ((pn )λ (π(A)) − fλ (π(A))) qα (π(A))λ(A) +pn (0)(qα )λ (π(A))λ(A) =  (pn (π(A)) − f (π(A)) − pn (0)) (π(A) − αI )−1 λ(A) +pn (0)(qα )λ (π(A))λ(A)   (pn (π(A)) − f (π(A))) (π(A) − αI )−1 λ(A)   +|pn(0)| (π(A) − αI )−1 λ(A) + (qα )λ (π(A))λ(A) → 0 as n → ∞.

Therefore lim pn qα − f qα  = 0.

n→∞

Furthermore, since {pn qα } ⊂ M, we obtain f qα ∈ M. The Proof of Theorem 2.3.2 Take an arbitrary f ∈ Cδ (Sp (A)). Since qα−1 f is continuous on Sp (A) \ {0}, we can choose a sequence {fn } of continuous functions on Sp (A) satisfying the following (1) fn (0) = 0, (2) fn (t) = (qα−1 f )(t), |t|  n1 , (3) sup |(fn qα )λ (t) − fλ (t)|  γ for some constant γ > 0. t ∈Sp (A)

Then we have sup |(fn qα )(t) − f (t)| =

t ∈Sp (A)

sup

|((fn qα )λ (t) − fλ (t))t|

− n1 t  n1

γ

1 → 0 as n → ∞. n

Furthermore, since (fn qα )λ converges pointwise to fλ and fλ μ < ∞, we have (fn qα )λ − fλ μ =

1 |(fn qα )λ (t) − fλ (t)|2 dμA (t)

2

→ 0 as n → ∞. Hence it follows that {fn qα } ⊂ M and limn→∞ fn qα − f  = 0, which implies f ∈ M. Thus M = Cδ (Sp (A)). By Lemma 2.3.3 and (2.3.10), we have {f (A); f ∈ Cδ (Sp (A))} = CT ∗ (A),

2.3 Functional Calculus for Self-adjoint Trio Observables

23

so that the map: f ∈ Cδ (Sp (A)) → f (A) ∈ CT ∗ (A) is surjective. Furthermore, using the functional calculus theorem for continuous functions of π(A) [Theorem A.1 in Appendix A], we can prove that the map: f → f (A) is a ∗-isomorphism of the Banach ∗-algebra Cδ (Sp (A)) onto the Banach ∗-algebra CT ∗ (A). Indeed, it is clear that the map f → f (A) is linear and ∗preserving. Furthermore, for any f, g ∈ Cδ (Sp (A)) it follows from (2.3.9) and (2.3.10) that   (fg)(A) = (fg)(π(A)), (fg)λ (π(A))λ(A), ((fg)∗λ (π(A))λ(A))∗   ¯ f¯λ (π(A))λ(A))∗ = f (π(A))g(π(A)), f (π(A))gλ (π(A))λ(A), (g(π(A)) = f (A)g(A),

and f  = max(f u , fλ μ ) = max(f (π(A)), fλ (π(A))λ(A)) = f (A). This completes the proof. In this paper we denote by [K] the closed subspace generated by a subset K of the Hilbert space H under the norm topology on H, and denote by Proj [K] the projection on the closed subspace [K] of H. Furthermore, we denote by ker X0 and R(X0 ) the kernel and the range of X0 ∈ B(H), respectively, namely ker X0 := {x ∈ H; X0 x = 0} and R(X0 ) := {X0 x; x ∈ H}. By Theorem 2.3.2 we get a decomposition theorem of a self-adjoint trio observable as follows: Corollary 2.3.6 Let A = A ∈ T ∗ (H) and PA the projection onto [R(π(A))]. We define elements As and An of T ∗ (H) by As = (π(A), PA λ(A), (PA λ(A))∗ ), An = (0, (I − PA )λ(A), ((I − PA )λ(A))∗ ). Then As , An ∈ CT ∗ (A) and A = As + An . Proof Define a function f on Sp (A) and a formal derivative f (0) at 0 as follows: f (t) = 0 on Sp (A) and f (0) = 1.

24

2 Fundamentals of Observable Algebras

Then,  fλ (t) = Let π(A) = and



0, t = 0 1, t = 0.

tdE(t) be the spectral resolution of π(A). Then since f (π(A)) = 0

fλ (π(A)) =

{0}

tdE(t)

= Proj ker π(A) = Proj R(π(A))⊥ = I − PA , where K ⊥ denotes the orthogonal complement of a subset K of H, it follows that f ∈ Cδ (Sp (A)), which implies by Theorem 2.3.2 that   f (A) = f (π(A)), fλ (π(A))λ(A), (f¯λ (π(A))λ(A))∗ = (0, (I − PA )λ(A), ((I − PA )λ(A))∗ ) = An ∈ CT ∗ (A). Hence, As = (π(A), PA λ(A), (PA λ(A))∗ ) = A − An ∈ CT ∗ (A). This completes the proof. The element As in Corollary 2.3.6 is called the semisimple part of A and An is called the nilpotent part of A. If A = As , then A is called semisimple, and if A = An , then A is called nilpotent. We next discuss a square root and a polar decomposition of A ∈ T ∗ (H) using Theorem 2.3.2. The following theorem shows that we can define the square root of A A. Let π(A) = UA (π(A)∗ π(A)) 2

1

be the polar decomposition H  of π(A). Here  UA is a unique partial  isometry on 1 1 such that UA∗ UA = Proj (π(A)∗ π(A)) 2 H and UA UA∗ = Proj (π(A)π(A)∗) 2 H (see Theorem A.2 in Appendix A). Then 1

π(A) = (π(A)π(A)∗ ) 2 UA . Using Theorem 2.3.2, we can prove a fundamental result in this note.

2.3 Functional Calculus for Self-adjoint Trio Observables

25 1

1

Theorem 2.3.7 Let A ∈ T ∗ (H). We define elements (AA ) 2 and (A A) 2 of T ∗ (H) by   1 1 (AA ) 2 = (π(A)π(A)∗ ) 2 , UA λ(A ), (UA λ(A ))∗ and   1 1 (A A) 2 = (π(A)∗ π(A)) 2 , UA∗ λ(A), (UA∗ λ(A))∗ . 1

Then they belong to CT ∗ (A) and they satisfy that ((AA ) 2 )2 = AA and 1 ((A A) 2 )2 = A A. Proof Putting 1

f (t) = t 2 ,

t  0,

we see that  fλ (t) =

1

t− 2 , t > 0 c, t = 0,

where c is any constant. Therefore f (π(AA )) = f (π(A)π(A)∗ ) = (π(A)π(A)∗ ) 2 1

and fλ (π(AA ))λ(AA ) = fλ (π(A)π(A)∗ )π(A)λ(A ) 1

= fλ (π(A)π(A)∗ )(π(A)π(A)∗ ) 2 UA λ(A ) = UA λ(A ), so that f ∈ Cδ (Sp (AA )), which implies by Theorem 2.3.2 that f (AA ) ∈ 1

CT ∗ (AA ), f (AA ) = (AA ) 2 and f (AA )2 = AA . Similarly, we can prove 1 1 that (A A) 2 ∈ CT ∗ (A) and ((A A) 2 )2 = A A. This completes the proof. 1

1

The elements (AA ) 2 and (A A) 2 of CT ∗ (A) in Theorem 2.3.7 are called the square roots of AA and A A, respectively. It is natural to consider the following 1 question: Is the square root (A A) 2 of A A unique? For this question we obtain the following

26

2 Fundamentals of Observable Algebras 1

Proposition 2.3.8 Let A ∈ T ∗ (H). Then, the square root (A A) 2 of A A is unique 1 1 (in the sense that if A A = B  B for B ∈ T ∗ (H), then (A A) 2 = (B  B) 2 ), if and only if ker π(A) = {0}. Proof Let A, B ∈ T ∗ (H). Then we have π(A)∗ π(A) = π(B)∗ π(B)

A A = B  B if and only if

and π(A)∗ λ(A) = π(B)∗ λ(B) if and only if

1

1

|π(A)|:=(π(A)∗π(A)) 2 =(π(B)∗ π(B)) 2 =|π(B)|

and |π(A)|UA∗ λ(A)=|π(B)|UB∗ λ(B)=|π(A)|UB∗ λ(B) |π(A)| = |π(B)|

if and only if and

UA∗ λ(A) − UB∗ λ(B)

∈ ker |π(A)|=ker π(A). (2.3.12)

Suppose that ker π(A) = {0} and A A = B  B for B ∈ T ∗ (H). Then it follows from Theorem 2.3.7 and (2.3.12) that   1 (A A) 2 = |π(A)|, UA∗ λ(A), (UA∗ λ(A))∗   = |π(B)|, UB∗ λ(B), (UB∗ λ(B))∗ 1

= (B  B) 2 . Suppose, conversely, that ker π(A) = {0}. Then take a non-zero vector x0 ∈ ker π(A) = ker |π(A)|, and define an element B of T ∗ (H) by   B = |π(A)|, UA∗ λ(A) + x0 , (UA∗ λ(A ) + x0 )∗ . Then we can prove that   B  B = π(A)∗ π(A), |π(A)|UA∗ λ(A), (|π(A)|UA∗ λ(A ))∗   = π(A)∗ π(A), π(A)∗ λ(A), (π(A)∗ λ(A ))∗ = A A, and that   1 (B  B) 2 = |π(A)|, UA∗ λ(A) + x0 , (UA∗ λ(A) + x0 )∗ 1

= (A A) 2 . 1

Thus the square root (A A) 2 of A A is not unique. This completes the proof.

2.3 Functional Calculus for Self-adjoint Trio Observables

27 1

Corollary 2.3.9 Let A = A ∈ T ∗ (H). If the square root (A A) 2 of A A is unique, then A is semisimple. Proof By Proposition 2.3.8 we have 1

(A A) 2 is unique if and only if ker π(A) = {0} if and only if PA = Proj [R(π(A))] = I which implies by Corollary 2.3.6 that A = As ,

namely, A is semisimple.

We next discuss the polar decomposition of A ∈ T ∗ (H). For A ∈ T ∗ (H) and B0 ∈ B(H) we define an element B0 A of T ∗ (H) by   B0 A = B0 π(A), B0 λ(A), (B0 λ(A ))∗ . Proposition 2.3.10 Let A ∈ T ∗ (H). Then the following statements hold. (1) Suppose that λ(A) ∈ [R(π(A))] and λ(A ) ∈ [R(π(A)∗ )]. Then 1

1

A + A = UA (A A) 2 + UA∗ (AA ) 2 . (2) Suppose that A = A , then 1

A is semisimple if and only if A = UA (A A) 2 . 1

In this case, this decomposition is unique, in the sense that if A = UB (B  B) 2 1 1 for some B ∈ T ∗ (H), then UB = UA and (B  B) 2 = (A A) 2 . Proof (1) Since UA UA∗ = Proj [R(π(A))] and UA∗ UA = Proj [R(π(A)∗ )], it follows that   1 UA (A A) 2 = UA |π(A)|, UA UA∗ λ(A), (UA UA∗ λ(A))∗ = (π(A), λ(A), λ(A)∗ ) and   1 UA∗ (AA ) 2 = UA∗ |π(A)∗ |, UA∗ UA λ(A ), (UA∗ UA λ(A ))∗ = (π(A)∗ , λ(A ), λ(A )∗ ).

28

2 Fundamentals of Observable Algebras

Therefore A + A = UA (A A) 2 + UA∗ (AA ) 2 . 1

1

(2.3.13)

(2) Suppose that A = A . Then, UA = UA∗ , and so by (2.3.13) we obtain 1

A = UA (A A) 2 .

(2.3.14)

1

Conversely, suppose that A = UA (A A) 2 . Then we have λ(A) = UA UA∗ λ(A) = PA λ(A), 1

1

which implies that A = As . Suppose that A = UA (A A) 2 = UB (B  B) 2 for some B ∈ T ∗ (H). Then we have π(A) = π(B) and λ(A) = UB UB∗ λ(B), which yields that UA = UB = UB∗ and λ(A) = PA λ(B).

(2.3.15)

Furthermore, since   UA (I − PA )λ(B)2 = UA2 (I − PA )λ(B)|(I − PA )λ(B) = 0, we have UA (I − PA )λ(B) = 0. Hence, it follows from (2.3.15) that UA λ(B) = UA PA λ(B) + UA (I − PA )λ(B) = UA λ(A), which implies that   1 (B  B) 2 = |π(B)|, UB∗ λ(B), (UB∗ λ(B  ))∗   = |π(A)|, UA λ(A), (UA λ(A))∗ 1

= (A A) 2 . Thus, the decomposition of A in (2.3.14) is unique; hence A is semisimple by Corollary 2.3.9. This completes the proof.

2.3 Functional Calculus for Self-adjoint Trio Observables

29

Finally, we consider a functional calculus of a general A ∈ T ∗ (H). For any polynomial: p(t) = α0 + α1 t + · · · + αn t n , we can check that p(AA ) ∈ T ∗ (H) if α0 = 0, and that p(AA )A = α0 A + α1 (AA )A + · · · + αn (AA )n A   ∗ = p(π(A)π(A)∗ )π(A), p(π(A)π(A)∗ )λ(A), (p(π(A) ¯ π(A))λ(A ))∗ ∈ T ∗ (H). Hence, for a general function f it is natural to write   f (AA )A= f (π(A)π(A)∗)π(A), f (π(A)π(A)∗ )λ(A), (f¯(π(A)∗ π(A))λ(A ))∗ . We consider that f (AA )A belongs to T ∗ (H) for what kind of functions f . Let  be a compact space and C() the set of all complex-valued continuous functions on . Then C() is a commutative C ∗ -algebra equipped with the usual function operations (f + g, αf, fg), the involution f → f ∗ = f¯ and the uniform norm f u := supt ∈ |f (t)|. We now construct a simple functional calculus of non-self adjoint trio observables which will be used in the proof of Theorem 2.3.12: Theorem 2.3.11 Let A ∈ T ∗ (H) and f ∈ C[0, π(A)2 ]. Then f (AA )A belongs to the closed subspace M(A) of CT ∗ (A) generated by {(AA )n A; n = 0, 1, . . .}, and f ∈ C[0, π(A)2 ] → f (AA )A ∈ M(A) is a continuous linear map. Proof It is clear that f (A A)A ∈ T ∗ (H) for any f ∈ C[0, π(A)2 ], and in particular it belongs to M(A) for any polynomial. Hence it follows from the Weierstrass approximation theorem that f (AA )A ∈ M(A) for any f ∈ C[0, π(A)2 ]. Furthermore, it is clear that f → f (AA )A is a continuous linear map of the Banach space C[0, π(A)2 ] into the Banach space M(A). This completes the proof. In Corollary 2.3.6 we have shown that every self-adjoint element A of T ∗ (H) is decomposed into the semisimple part As ∈ CT ∗ (A) and the nilpotent part An ∈ CT ∗ (A). Using Theorem 2.3.11, we show that this decomposition is possible for a general element of T ∗ (H). For A ∈ T ∗ (H), write P (CT ∗ (A)) = the CT ∗ - algebra generated by {ST ; S, T ∈ CT ∗ (A)}, N(CT ∗ (A)) = {T ∈ CT ∗ (A); π(T ) = 0}.

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2 Fundamentals of Observable Algebras

We now define elements As and An of T ∗ (H) as follows:   As = π(A), PA λ(A), (PA λ(A ))∗ ,   An = 0, (I − PA )λ(A), ((I − PA )λ(A ))∗ . Theorem 2.3.12 Every element A of T ∗ (H) is decomposed into: A = As + An , where As ∈ P (CT ∗ (A)) and An ∈ N(CT ∗ (A)). Proof Put  χ0 (t) =

1, t > 0 0, t = 0

and χ0 (AA )A   = χ0 (π(A)π(A)∗ )π(A), χ0 (π(A)π(A)∗)λ(A), (χ0 (π(A)∗ π(A))λ(A ))∗ . Let |π(A)∗ | = tion:

 π(A) 0

tdE(t) be the spectral resolution of |π(A)∗ |. The calcula-

∗

χ0 (π(A)π(A) ) =

π(A)

χ0 (t 2 )dE(t)

0

= I − E(0) = I − Proj ker |π(A)∗ | = I − Proj R(π(A))⊥ = PA ,

(2.3.16)

implies that   χ0 (AA )A = PA π(A), PA λ(A), (PA λ(A ))∗   = π(A), PA λ(A), (PA λ(A ))∗ = As .

(2.3.17)

2.3 Functional Calculus for Self-adjoint Trio Observables

31

Now, consider a sequence {fn } of continuous functions on R+ :  fn (t) =

1, t > n1 nt, 0  t  n1 .

By Theorem 2.3.11 we have fn (AA )A ∈ M(A) ⊂ P (CT ∗ (A)).

(2.3.18)

Since √ √ 1 sup |fn (t) t − χ0 (t) t|  , n t ∈R+

n ∈ N,

it follows from (2.3.16) that lim fn (π(A)π(A)∗ )π(A) − χ0 (π(A)π(A)∗ )π(A)

n→∞

= lim fn (π(A)π(A)∗ )π(A) − PA π(A) n→∞

= 0. Furthermore, since limt →∞ fn (t) = χ0 (t), t ∈ R+ , it follows that lim fn (π(A)π(A)∗ )λ(A) − χ0 (π(A)π(A)∗ )λ(A)

n→∞

= lim fn (π(A)π(A)∗ )λ(A) − PA λ(A) n→∞

= 0, and that lim fn (π(A)∗ π(A))λ(A ) − PA λ(A ) = 0,

n→∞

which implies by (2.3.17) that lim fn (AA )A − As  = 0.

n→∞

Hence it follows from (2.3.18) that As ∈ P (CT ∗ (A)), so that An = A − As ∈ N(CT ∗ (A)). This completes the proof. The element As of P (CT ∗ (A)) in Theorem 2.3.12 is called the semisimple part of A, and the element An of N(CT ∗ (A)) is called the nilpotent part of A. If A = As , then A is called semisimple, and if A = An , then it is called nilpotent.

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2 Fundamentals of Observable Algebras

Notes For the spectrum, the spectral resolution, the functional calculus and the polar decomposition of a bounded linear operator in Hilbert space used in this section, refer to Appendix A.

2.4 ∗-Automorphisms of Observable Algebras To investigate ∗-automorphisms of T ∗ (H), we first define three fundamental ∗automorphisms of T ∗ (H). For any unitary operator U on H, α ∈ C and g ∈ H we put IU (A) = (U π(A)U ∗ , U λ(A), (U λ(A ))∗ ), Sα (A) = (π(A), αλ(A), (αλ(A ))∗ ), Tg (A) = (π(A), π(A)g + λ(A), (π(A)∗ g + λ(A ))∗ ) for A ∈ T ∗ (H). Then it is easily shown that IU , Sα and Tg are continuous ∗automorphisms of the CT ∗ -algebra T ∗ (H). They are called the unitary transform, the similar transform and the translation of T ∗ (H), respectively. The following theorem is the main result of this section: Theorem 2.4.1 For any ∗-automorphism σ of the CT ∗ -algebra T ∗ (H) there exist a unitary operator U on H, α ∈ C and g ∈ H such that σ = Tg ◦ IU ◦ Sα . Proof Put σ0 (π(A)) = π(σ (A)), A ∈ T ∗ (H). Suppose π(A) = 0. Then, since A A = 0, we have σ (A) σ (A) = σ (A A) = 0, which implies that π(σ (A)) = 0. Hence σ0 is well-defined, and it is easily shown that σ0 is a ∗-automorphism of B(H). Then there exists a unitary operator U on H such that σ0 (π(A)) = U π(A)U ∗ ,

A ∈ T ∗ (H)

(2.4.1)

(see Notes of this section). Here, for any X0 ∈ B(H) and x ∈ H we define elements X˜ 0 and x˜ of T ∗ (H) by X˜ 0 = (X0 , 0, 0), x˜ = (0, x, 0).

2.4 ∗-Automorphisms of Observable Algebras

33

By (2.4.1) there exist g and h of H such that σ (I˜) = (U I U ∗ , g, h∗ ) = (I, g, h∗ ). Since I˜ = I˜, we have σ (I˜) = σ (I˜ ) = (I, h, g ∗ ), hence, g = h and σ (I˜) = (I, g, g ∗ ).

(2.4.2)

By (2.4.1) again, there exist element ξ and η of H such that σ (X˜ 0 ) = (U X0 U ∗ , ξ, η∗ ). Since X˜ 0 I˜ = X˜ 0 and I˜X˜ 0 = X˜ 0 , we have σ (X˜ 0 ) = σ (X˜ 0 I˜) = σ (X˜ 0 )σ (I˜) = (U X0 U ∗ , U X0 U ∗ g, η∗ ) and σ (X˜ 0 ) = σ (I˜)σ (X˜ 0 ) = (U X0 U ∗ , ξ, (U X0∗ U ∗ g)∗ ), which implies by (2.4.2) that for X0 ∈ B(H), σ (X˜ 0 ) = (U X0 U ∗ , U X0 U ∗ g, (U X0 U ∗ g)∗ )

(2.4.3)

(T−g ◦ σ )(X˜ 0 ) = (U X0 U ∗ , 0, 0).

(2.4.4)

and

Now, put σ = IU ∗ ◦ T−g ◦ σ. By (2.4.4) we obtain σ (X˜ 0 ) = X˜ 0 , X0 ∈ B(H).

(2.4.5)

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2 Fundamentals of Observable Algebras

Let x ∈ H. Since σ (x˜  )σ (x) ˜ = σ (x˜  x) ˜ = O, there exist ξx and ηx of H such that ˜ = (0, ξx , ηx∗ ), σ (x) so that by (2.4.2), ˜ = σ (I˜)σ (x) ˜ = (0, ξx , 0). σ (x)

(2.4.6)

We here define a linear map  of H onto H by x = ξx , x ∈ H. Then it follows from (2.4.5) and (2.4.6) that ˜ (0, X0 x, 0) = σ ((0, X0 x, 0)) = σ (X˜ 0 x) ˜ = X˜ 0 σ (x) ˜ = σ (X˜ 0 )σ (x) = (0, X0 x, 0) for all X0 ∈ B(H) and x ∈ H, which implies that X0 = X0  for all X0 ∈ B(H). Though we don’t know whether  is bounded or not, we can show that  = αI for some α ∈ C. Indeed, since (x|η)ξ = (ξ ⊗ η)x ¯ = (ξ ⊗ η)x ¯ = (x|η)ξ for all x, ξ, η ∈ H, where (ξ ⊗ η)x ¯ := (x|η)ξ , it follows that (x|η) = (ξ |ξ )(x|η) for all x, η ∈ H and ξ ∈ H1 := {ζ ∈ H; ζ  = 1}. Furthermore, we can prove that (ξ |ξ ) does not depend on the method of taking ξ ∈ H1 . Hence,  = αI for some α ∈ C, so that σ ((0, x, 0)) = (0, αx, 0) and σ ((0, 0, y ∗ )) = σ ((0, y, 0) ) = σ ((0, y, 0)) = (0, αy, 0) = (0, 0, (αy)∗ )

2.4 ∗-Automorphisms of Observable Algebras

35

for all x, y ∈ H. Thus we have σ ((0, x, y ∗ )) = (0, αx, (αy)∗ ). Therefore, it follows from (2.4.5) that σ (X) = X˜ 0 + (0, αx, (αy)∗ ) = (X0 , αx, (αy)∗ ) = Sα (X) for all X ∈ T ∗ (H), which yields that σ = Tg ◦ IU ◦ Sα . This completes the proof. By Theorem 2.4.1 every ∗-automorphism of the CT ∗ -algebra T ∗ (H) always continuous. We next discuss ∗-automorphisms of Q∗ (H). Put τ (A) = (π(A), λ(A), λ(A )∗ ) for A = (π(A), λ(A), λ(A )∗ , μ(A)) ∈ Q∗ (H). Then, τ (Q∗ (H)) = T ∗ (H). For any unitary operator U on H, α ∈ C and g ∈ H we define continuous ∗q q q automorphisms IU (A), Sα (A) and Tg (A) of the Banach ∗-algebra Q∗ (H) by q

IU (A) = (IU (τ (A)), μ(A)), Sαq (A) = (Sα (τ (A)), |α|2 μ(A)),   q Tg (A) = Tg (τ (A)), (μ(A) + (λ(A)|g) + (g|λ(A )) + (π(A)g|g)) for A ∈ Q∗ (H), which are called the unitary transform, the similar transform and the translation of Q∗ (H), respectively. Lemma 2.4.2 Suppose that σ is a ∗-automorphism of the CQ∗ -algebra Q∗ (H). Then there exist a unitary operator U on H, α ∈ C and g ∈ H such that q

q

σ (AB) = (Tg ◦ IU ◦ Sαq )(AB) for all A, B ∈ Q∗ (H).

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2 Fundamentals of Observable Algebras

Proof Suppose τ (A) = O. Then, A = (0, 0, 0, μ(A)) and A = (0, 0, 0, μ(A)). ¯ Let σ (A) = (X0 , x, y, α). Since A A = AA = O, we have O = σ (A A) = σ (A) σ (A) = (X0∗ X0 , X0∗ x, (X0∗ x)∗ , x2 ), O = σ (AA ) = (X0 X0∗ , X0 y, (X0 y)∗ , y2 ), so that X0 = 0, x = y = 0. Hence τ (σ (A)) = O. Thus we can define a ∗-automorphism στ of the CT ∗ -algebra T ∗ (H) by στ (τ (A)) = τ (σ (A)), A ∈ Q∗ (H). By Theorem 2.4.1 there exist a unitary operator U on H, α ∈ C and g ∈ H such that στ = Tg ◦ IU ◦ Sα ,

(2.4.7)

so that π(σ (A)) = U π(A)U ∗ , λ(σ (A)) = U π(A)U ∗ g + αU λ(A),

(2.4.8)

λ(σ (A) ) = U π(A)∗ U ∗ g + αU λ(A )

(2.4.9)

for all A ∈ Q∗ (H). Hence it follows from (2.4.8) and (2.4.9) that μ(σ (AB)) = μ(σ (A)σ (B)) = (λ(σ (B))|λ(σ (A) )) = (U π(B)U ∗ g + αU λ(B)|U π(A)∗ U ∗ g + αU λ(A )) = (U π(AB)U ∗ g|g) + (g|αU λ(B  A )) +(αU λ(AB)|g) + |α|2 μ(AB). On the other hand, since Sαq (AB) = (Sα (τ (AB)), |α|2 μ(AB)), q

(IU ◦ Sαq )(AB) = (IU ◦ Sα (τ (AB)), |α|2 μ(AB)), we have q

q

q

q

μ((Tg ◦ IU ◦ Sαq )(AB)) = μ((IU ◦ Sαq )(AB)) + (λ((IU ◦ Sαq )(AB)|g) q

q

+(g|λ((IU ◦ Sαq )(AB) )) + (π(IU ◦ Sαq )(AB)g|g) = α 2 μ(AB) + (αU λ(AB)|g) + (g|αU λ((AB) )) +(U π(AB)U ∗ g|g) = μ(σ (AB)),

2.4 ∗-Automorphisms of Observable Algebras

37

which implies by (2.4.7) that σ (AB) = (στ (τ (AB)), μ(σ (AB))) q

q

= ((Tg ◦ IU ◦ Sα )τ (AB), μ((Tg ◦ IU ◦ Sαq )(AB))) q

q

= (Tg ◦ IU ◦ Sαq )(AB). This completes the proof. By Lemma 2.4.2 we obtain the following result for ∗-automorphisms of the CQ∗ -algebra Q∗ (H): Theorem 2.4.3 Suppose that σ is a ∗-automorphism of the CQ∗ -algebra Q∗ (H). Then there exist a unitary operator U on H, α ∈ C and g ∈ H such that q

q

σ (A) = (Tg ◦ IU ◦ Sαq )(A) for all A ∈ Q∗ (H), that is, π(σ (A)) = U π(A)U ∗ , λ(σ (A)) = (U π(A)U ∗ )g + αU λ(A), λ∗ (σ (A)) = ((U π(A)∗ U ∗ )g + αU λ(A ))∗ , μ(σ (A)) = |α|2 μ(A) + (αU λ(A)|g) + (g|αU λ(A )) +(U π(A)U ∗ g|g). Proof Take an arbitrary A ∈ Q∗ (H). Then we define elements I˜, B, C of Q∗ (H) by I˜ = (I, 0, 0, 0),   1 1 1 π(A), λ(A), λ(A )∗ , μ(A) , B= 2 2 2   1 1 1 π(A), λ(A), λ(A )∗ , μ(A) . C= 2 2 2 Then, by the equality: A = B I˜ + I˜C + μ(A)V , we get σ (A) = σ (B I˜) + σ (I˜C) + μ(A)σ (V ).

(2.4.10)

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2 Fundamentals of Observable Algebras

By Lemma 2.4.2, we obtain   1 σ (B I˜) = (Tg ◦ IU ◦ Sα )(τ (B I˜)), (g|αU λ(A )) + (U π(A)U ∗ g|g) 2 (2.4.11) and   1 ∗ ˜ ˜ σ (I C) = (Tg ◦ IU ◦ Sα )(τ (I B)), (αU λ(A)|g) + (U π(A)U g|g) . 2 (2.4.12) For any x ∈ H with x = 1, define an element x˜ of Q∗ (H) by x˜ = (0, x, 0, 0). ˜ = σ (V ). Then, x˜  = (0, 0, x ∗ , 0) and x˜  x˜ = (0, 0, 0, x2 ) = V . Hence, σ (x˜  x) On the other hand, by Lemma 2.4.2 σ (x˜  x) ˜ = (τ (σ (x˜  x)), ˜ |α|2 ) = |α|2 V . Thus, σ (V ) = |α|2 V .

(2.4.13)

By (2.4.10)–(2.4.13) we have τ (σ (A)) = (Tg ◦ IU ◦ Sα )(τ (A)), μ(σ (A)) = (g|αU λ(A )) + (U π(A)U ∗ g|g) + |α|2 μ(A) q

q

= μ((Tg ◦ IU ◦ Sαq )(A)), which completes the proof. By Theorem 2.4.3 every ∗-automorphism of Q∗ (H) is continuous. Notes Every ∗-automorphism α of B(H) is represented as α(A) = U AU ∗ , A ∈ B(H) for some unitary operator U on H. Indeed, for any ξ, η ∈ H we denote by ξ ⊗ η¯ the one dimensional operator on H by (ξ ⊗ η)x ¯ = (x|η)ξ, x ∈ H.

2.5 Locally Convex Topologies on T ∗ -algebras

39

Let x0 ∈ H with x0  = 1. Then it is easily shown that x0 ⊗ x¯0 is a non-zero projection on H. Hence, α(x0 ⊗ x¯0 ) is a non-zero projection on H. Here, take an element ξ0 of H such that ξ0  = 1 and α(x0 ⊗ x¯0 )ξ0 = ξ0 . Then we have (x0 ⊗ x¯0 )A(x0 ⊗ x¯0 ) = (Ax0 |x0 )(x0 ⊗ x¯0 ) for all A ∈ B(H), which implies that (α(A)ξ0 |ξ0 ) = (α((x0 ⊗ x¯0 )A(x0 ⊗ x¯0 ))ξ0 |ξ0 ) = (Ax0 |x0 ). Furthermore, since B(H)x = α(B(H))x = H for any non-zero x ∈ H, we can define a unitary operator U on H by U Ax0 = α(A)ξ0 . Hence, U AU ∗ ξ0 = α(A)ξ0 for all A ∈ B(H). Take an arbitrary x ∈ H. Then there exists an element B of B(H) such that x = α(B)ξ0 = U Bx0 , which yields that U AU ∗ x = U ABξ0 = α(AB)ξ0 = α(A)x. Thus, α(A) = U AU ∗ for all A ∈ B(H). For the spatiality of ∗-automorphisms of more general von Neumann algebras, refer to [35].

2.5 Locally Convex Topologies on T ∗ -algebras As mentioned in Sect. 2.2, T ∗ (H) is a Banach ∗-algebra with the norm: A = max(π(A), λ(A), λ(A )), A ∈ T ∗ (H). The topology defined by the norm  ·  is called the uniform topology and is denoted by τu . We define other locally convex topologies on T ∗ (H). The product topology on T ∗ (H) = B(H) × H × H ∗ defined in terms of the weak (resp. the σ -weak) topology on B(H) and the weak topology on H and H ∗ is called the weak (resp. the σ -weak) topology on T ∗ (H) and is denoted by τw (resp. τσ w ). The product topology on T ∗ (H) defined in terms of the strong (resp. the σ -strong, strong∗ and σ -strong∗) topology on B(H) (see Appendix C.3) and the norm-topology on H and H ∗ is called the strong (resp. the σ -strong, strong∗ and σ - strong∗ ) topology on T ∗ (H) and is denoted by τs (resp.

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2 Fundamentals of Observable Algebras

τσ s , τs∗ and τσ∗s ). To be concrete, the σ -strong∗ topology τσ∗s on T ∗ (H) is defined by a family {px∗ ; x = {xn } ∈ HN } of seminorms: px∗ (A)

=

∞ 

π(A)xn  + 2

n=1

∞ 

1 ∗

2

π(A) xn  + λ(A) + λ(A ) 2

2



2

,

n=1

and the σ -weak topology τσ w is defined by a family {px,y,F,G ; x = {xn }, y = {yn } ∈ HN , F, G ∈ F} of seminorms: for F = {f1 , . . . , fl }, G = {g1 , . . . , gm } ∈ F, ⎡ ⎤ ∞ l m    px,y,F,G (A) = ⎣| (π(A)xn |yn )| + | (λ(A)|fk )| + | (gj |λ(A )|⎦ , n=1

k=1

j =1

 2 where HN := {{xn } ⊂ H; ∞ n=1 xn  < ∞}, and F is the set of all finite subsets of ∗ H. Similarly, other topologies τs , τs , τσ s and τw are concretely defined by families of seminorms. As for their topologies we have the following





τw ≺ τs ≺ τσ s (2.5.1)

τs∗ ≺ τσ∗s ≺ τu , where τ1 ≺ τ2 means that a topology τ2 is finer than a topology τ1 . Let τ be one of the above topologies on T ∗ (H). We denote T ∗ (H) equipped with τ by T ∗ (H)[τ ], and denote the induced topology of T ∗ (H)[τ ] on a subset A ¯ ]. Then we can of T ∗ (H) by A[τ ]. We denote the closure of A[τ ] in T ∗ (H) by A[τ easily show the following Proposition 2.5.2 Let A be a T ∗ -algebra on H. Then A[τw ], A[τσ w ], A[τs∗ ] and A[τσ∗s ] are locally convex ∗-algebras. In particular, T ∗ (H)[τs∗ ] and T ∗ (H)[τσ∗s ] are complete locally convex ∗-algebras. For the definition of locally convex (∗)-algebras refer to Appendix C.4. A[τs ] and A[τσ s ] are not necessarily locally convex algebras. Indeed, suppose that a net {Aα } in A converges strongly to an element A of A. For an element B of A we have Aα B → AB, strongly, but BAα does not necessarily converges strongly to BA because π(Aα )∗ λ(B) does not necessarily converge to π(A)∗ λ(B) under the norm topology on H. Hence the multiplication of A[τs ] is not separately continuous, which means that A[τs ] is not a locally convex algebra in general. Similarly, A[τσ s ] is not a locally convex algebra in general. We next consider the closures of a T ∗ -algebra A for their topologies. Since T ∗ (H)[τs ], T ∗ (H)[τσ s ], T ∗ (H)[τs∗ ] and ¯ s ], A[τ ¯ σ s ], A[τ ¯ ∗ ] and A[τ ¯ ∗ ] of A under T ∗ (H)[τσ∗s ] are complete, the closures A[τ s σs ∗ ∗ the strong, σ -strong, strong and σ -strong topologies, respectively are contained in T ∗ (H), but the completions of A under the weak topology and the σ -weak topology

2.6 Commutants and Bicommutants of T ∗ -algebras

41

¯ w ] and A[τ ¯ σ w ] the are not necessarily contained in T ∗ (H), so that we denote by A[τ closure of A in T ∗ (H) under the weak and the σ -weak topologies, respectively. Now we have the following Proposition 2.5.3 Let A be a T ∗ -algebra on H. Then, all of the closures of A under the topologies τw , τσ w , τs , τσ s , τs∗ and τσ∗s coincide. Proof By (2.5.1) we have

⊂

⊂

¯ s ] ⊂ A[τ ¯ σ w ] ⊂ A[τ ¯ w] ¯ σ s ] ⊂ A[τ A[τ ¯ s∗ ] ¯ σ∗s ] ⊂ A[τ A[τ

(2.5.2)

and by the von Neumann type density theorem [Theorem 3.1.1] in Sect. 3.1 ¯ w ]. ¯ ∗ ] = A[τ A[τ σs Hence, all of their closures of A coincide. This completes the proof.

2.6 Commutants and Bicommutants of T ∗ -algebras In this section we define and investigate three commutants Aπ , Aτ and Aρ , and three bicommutants Aππ , Aτ τ and Aρρ of a T ∗ -algebra A. We first define commutants of T ∗ -algebras. Definition 2.6.1 Let A be a T ∗ -algebra on H. We define the commutants Aπ , Aτ and Aρ of A as follows: Aπ = {K ∈ T ∗ (H); π(K) ∈ π(A) (the commutant of π(A))}, Aτ = {K ∈ T ∗ (H); AK = KA for all A ∈ A}, Aρ = {K ∈ Aτ ;

(λ(A)|λ(K  )) = (λ(K)|λ(A )) for all A ∈ A}.

Then we have the following Proposition 2.6.2 The commutants Aπ , Aτ and Aρ of A are CT ∗ -algebras which are closed under all of τw , τσ w , τs , τσ s , τs∗ and τσ∗s topologies and (Aτ )2 ⊂ Aρ ⊂ Aτ ⊂ Aπ .

(2.6.1)

The commutants Aρ , Aτ and Aπ don’t coincide in general. Proof It is easily shown that the commutants Aπ , Aτ and Aρ of A are CT ∗ -algebras satisfying (2.6.1). Suppose now that a net {Kα } in Aτ converges weakly to an

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2 Fundamentals of Observable Algebras

element K of T ∗ (H). Then, for any A ∈ A, Kα A − → KA, weakly, that is, α

→ π(K)π(A), weakly, π(Kα )π(A) − α

→ π(K)λ(A), weakly, π(Kα )λ(A) − α

π(A)∗ λ(Kα ) − → π(A)∗ λ(K  ), weakly α

→ AK, weakly. Because Kα A = AKα , we get KA = AK, and similarly, AKα − α

and so K ∈ Aτ . Thus, Aτ is weakly closed. It is similarly shown that Aπ and Aρ are also weakly closed. In Example 2.6.4 later, it will see that each of their commutants does not coincide in general. This completes the proof. As for the relationship between the fundamental ∗-automorphisms of T ∗ (H) and the commutants of CT ∗ -algebras, we get the following Proposition 2.6.3 Let A be a CT ∗ -algebra on H and σ a fundamental ∗automorphism of T ∗ (H). Then, σ (A)π = σ (Aπ ), σ (A)τ = σ (Aτ ) and σ (A)ρ = σ (Aρ ). Proof Let σ = Tg , g ∈ H. The equality: Tg (A) = (π(A), π(A)g + λ(A), ((π(A))∗ g + λ(A ))∗ ), A ∈ A implies that Tg (A)π = Tg (Aπ ).

(2.6.2)

Take an arbitrary K ∈ Tg (A)τ . Then we can calculate that π(K)π(A) = π(A)π(K), π(K)(π(A)g + λ(A)) = π(A)λ(K), π(K)∗ (π(A)∗ g + λ(A )) = π(A)∗ λ(K  ) for all A ∈ A, so that π(K)λ(A) = π(A)(λ(K) − π(K)g), π(K)∗ λ(A ) = π(A)∗ (λ(K  ) − π(K)∗ g). Therefore, X := T−g (K) ∈ Aτ and K = Tg (X) ∈ Tg (Aτ ). Thus, Tg (A)τ ⊂ Tg (Aτ ).

(2.6.3)

2.6 Commutants and Bicommutants of T ∗ -algebras

43

Conversely suppose K ∈ Tg (A)ρ . Since Tg (A)ρ ⊂ Tg (A)τ ⊂ Tg (Aτ ) by (2.6.3), there exists an element X of Aτ such that K = Tg (X), which implies that (λ(K)|λ(Tg (A) )) = (λ(X) + π(X)g|π(A)∗ g + λ(A )) = (λ(X)|π(A)∗ g) + (λ(X)|λ(A )) +(π(X)g|π(A)∗ g) + (π(X)g|λ(A )) = (λ(A)|π(X)∗ g) + (λ(X)|λ(A )) +(π(A)g|π(X)∗ g) + (π(A)g|λ(X )) and that (λ(Tg (A))|λ(K  )) = (π(A)g|λ(X )) + (π(A)g|π(X)∗ g) +(λ(A)|λ(X )) + (λ(A)|π(X)∗ g) for all A ∈ A. Hence we obtain (λ(X)|λ(A )) = (λ(A)|λ(X )) for all A ∈ A. Therefore, X ∈ Aρ and K = Tg (X) ∈ Tg (Aρ ). Thus, Tg (A)ρ ⊂ Tg (Aρ ).

(2.6.4)

The converses of (2.6.3) and (2.6.4): Tg (Aτ ) ⊂ Tg (A)τ and Tg (Aρ ) ⊂ Tg (A)ρ are easily shown. The cases of σ = IU and σ = Sα are proved in the same way as the case σ = Tg . This completes the proof. We next we discuss the bicommutants Aππ := (Aπ )π , Aτ τ := (Aτ )τ and Aρρ := (Aρ )ρ of A. It is immediate that Aππ = {A ∈ T ∗ (H); π(A) ∈ π(A) }, A ⊂ AT := Aππ ∩ Aτ τ ∩ Aρρ , but the bicommutants Aππ , Aτ τ and Aρρ don’t have mutual relations in general as seen in next Example 2.6.4. In Sect. 3.1, we will show that A[τσ∗s ] = AT for every CT ∗ -algebra A [the von Neumann type density theorem]. We here give commutants and bicommutants of special T ∗ -algebras.

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2 Fundamentals of Observable Algebras

Example 2.6.4 (1) Let T ∗ (H) = (B(H), H, H ∗ ). Then T ∗ (H)π = (CI, H, H ∗ ), T ∗ (H)τ = T ∗ (H)ρ = {O}, and AT = T ∗ (H) = T ∗ (H)ππ = T ∗ (H)τ τ = T ∗ (H)ρρ . (2) Let A = (CI, H, H ∗ ). Then Aπ = T ∗ (H)  Aτ = Aρ = {O}, and ττ ρρ ∗ AT = A = Aππ ⊂ = A = A = T (H).

(3) Let A = (0, H, H ∗ ). Then we have Aπ = T ∗ (H)  Aτ = A  Aρ = {O} Aππ = (CI, H, H ∗ ),

Aτ τ = A, Aρρ = T ∗ (H),

so that ππ ⊂ ρρ AT = A = Aτ τ ⊂ = A = A .

(4) Let A0 be a nondegenerate C ∗ -algebra on H, and g a non-zero element of H. We define a CT ∗ -algebra A by {(A0 , A0 g, (A0 g)∗ ); A0 ∈ A0 }. Then we see that Aπ  Aτ = Aρ = {(K0 , K0 g, (K0∗ g)∗ ); K0 ∈ A 0 }, ττ ρρ ∗ ∗ Aππ = (A 0 , H, H ∗ ) ⊃ = A = A = {(A0 , A0 g, (A0 g) );

Thus ¯ ∗ ]  Aππ . AT = Aτ τ = Aρρ = A[τ σs Putting ν(X) = (λ(X), λ(X )∗ ),

X ∈ T ∗ (H),

A0 ∈ A 0 }.

2.6 Commutants and Bicommutants of T ∗ -algebras

45

ν(T ∗ (H)) is a Hilbert space with inner product: (ν(X)|ν(Y )) = (λ(X)|λ(Y )) + (λ(Y  )|λ(X )) for X, Y ∈ T ∗ (H), which is identical with the direct sum H ⊕ H ∗ of the Hilbert spaces H and H ∗ . The subspaces ν(A) and ν(Aρρ ) of the Hilbert space ν(T ∗ (H)) have the following property: Proposition 2.6.5 Let A be a CT ∗ -algebra on H. Then ν(A) is dense in ν(Aρρ ), that is, ν(Aρρ ) ⊂ [ν(A)], where [ν(A)] is the closure of ν(A) in the Hilbert space ν(T ∗ (H)) under the norm topology. This will be used in the proof of the von Neumann type density theorem in Chap. 3. In order to prove Proposition 2.6.5 we prepare some lemmas. Lemma 2.6.6 Let [λ(A)] be the closure of λ(A) in the Hilbert space H under the norm topology. Then λ(Aρρ ) ⊂ [λ(A)]. Proof Let CA be the projection on [λ(A)]. Since π(A)[λ(A)] ⊂ [λ(A)], we have CA ∈ π(A) , and so (I − CA )∼ := (I − CA , 0, 0) ∈ Aρ . Hence, since (I − CA )λ(A) = π(A)λ((I − CA )∼ ) = 0 for all A ∈ Aρρ , it follows that CA λ(A) = λ(A) for all A ∈ Aρρ . Therefore, λ(A) ∈ [λ(A)] for all A ∈ Aρρ . This completes the proof. Lemma 2.6.7 Let J be an Hermitian unitary operator on the Hilbert space ν(T ∗ (H)) defined by 

I 0 0 −I

 .

Then, ν(Aρ ) ⊂ J ν(A)⊥ and ν(Aρρ ) ⊂ J ν(Aρ )⊥ , where ν(A)⊥ and ν(Aρ )⊥ are orthogonal complements of ν(A) and ν(Aρ ) in the Hilbert space ν(T ∗ (H)), respectively.

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2 Fundamentals of Observable Algebras

Proof Take an arbitrary K ∈ Aρ . Then we have   (J ν(K)|ν(A)) = (λ(K), −λ(K  )∗ )|(λ(A), λ(A )∗ ) = (λ(K)|λ(A)) − (λ(A )|λ(K  )) =0 for all A ∈ A. Hence, ν(Aρ ) ⊂ J ν(A)⊥ . Applying the same argument to Aρ instead of A, we can show that ν(Aρρ ) ⊂ J ν(Aρ )⊥ . Lemma 2.6.8 Let H0 = [λ(A)] ⊕ [λ∗ (A)]. Then ν(Aρ ) ∩ H0 is dense in J ν(A)⊥ ∩ H0 . Proof Take an arbitrary (x, y ∗ ) ∈ J ν(A)⊥ ∩ H0 , that is, x, y ∈ [λ(A)] and (λ(A)|y) = (x|λ(A ))

(2.6.5)

for all A ∈ A. We now define operators X0 and Y0 on λ(A) by X0 λ(A) = π(A)x, , Y0 λ(A) = π(A)y, A ∈ A.

(2.6.6)

Then it follows from (2.6.5) that (λ(B)|Y0 λ(A)) = (λ(A B)|y) = (x|λ(B  A)) = (X0 λ(B)|λ(A)) for all A, B ∈ A. Furthermore, since the closed subspace [λ(A)] of H is invariant under the action π(A), X0 and Y0 are well-defined closable operators in the Hilbert space [λ(A)] satisfying Y0 ⊂ X0∗ and X0 ⊂ Y0∗ .

(2.6.7)

We denote by X¯ 0 and Y¯0 their closures of X0 and Y0 in the Hilbert space [λ(A)], respectively. For closable operators and closed operators in a Hilbert space refer to Appendix B.1. Using the projection CA := Proj [λ(A)] in π(A) , we can define closed operators X¯ 0 CA and Y¯0 CA in the Hilbert space H. Then we see that X¯ 0 CA = CA X¯ 0 CA and Y¯0 CA = CA Y¯0 CA

(2.6.8)

and X¯ 0 CA and Y¯0 CA are affiliated with

π(A) ,

(2.6.9)

2.6 Commutants and Bicommutants of T ∗ -algebras

47

namely A0 (X¯ 0 CA ) ⊂ (X¯ 0 CA )A0 and A0 (Y¯0 CA ) ⊂ (Y¯0 CA )A0 for all A0 ∈ π(A) . Indeed, (2.6.8) is trivial. We will show (2.6.9). It follows immediately from (2.6.6) that π(A)(X¯ 0 CA ) ⊂ (X¯ 0 CA )π(A) for all A ∈ A and B0 (X¯ 0 CA ) ⊂ (X¯ 0 CA )B0 for all B0 ∈ π(A)[τsπ ] (the strong closure of π(A)), which implies by the von Neumann density theorem (Theorem C.1 in Appendix C) that A0 (X¯ 0 CA ) ⊂ (X¯ 0 CA )A0 for all A0 ∈ π(A) . Similarly, A0 (Y¯0 CA ) ⊂ (Y¯0 CA )A0 for all A0 ∈ π(A) . Furthermore, it follows from (2.6.7) that Y¯0 CA = CA Y¯0 CA ⊂ CA X0∗ CA ⊂ (CA X¯ 0 CA )∗ = (X¯ 0 CA )∗ .

(2.6.10)

Now, write X = (X¯ 0 CA , x, y ∗ ) and π(X) = X¯ 0 CA , λ(X) = x and λ(X ) = y. By (2.6.9) and (2.6.10) we have π(X)π(A) ⊃ π(A)π(X), π(X)λ(A) = X¯ 0 CA λ(A) = π(A)x = π(A)λ(X), π(A)∗ λ(X ) = π(A)∗ y = (Y¯0 CA )λ(A ) = (X¯ 0 CA )∗ λ(A ) = π(X)∗ λ(A )

(2.6.11)

for all A ∈ A. This means that X may regard as an unbounded trio-observable in H which commutes with the T ∗ -algebra A. Using the polar decomposition of π(X) and the spectral resolutions of π(X)π(X)∗ and π(X)∗ π(X), we can find a sequence {Xn } in Aρ satisfying limn→∞ ν(Xn ) = (x, y ∗ ). Let π(X) = U |π(X)| be the polar ∗ ∗ decomposition of π(X). Then, π(X)∗ = decomposition of  ∞U |π(X) | is the polar ∗ 2 ∗ π(X) . Let |π(X)| = π(X) π(X) = 0 tdE(t) and |π(X)∗ |2 = π(X)π(X)∗ = ∞ 2 ∗ 2 0 tdF (t) be the spectral resolutions of |π(X)| and |π(X) | , respectively. For the polar decomposition of a densely defined closed operator and the spectral resolution of a self-adjoint operator, see Theorems B.10 and B.5 in Appendix B, respectively. Then we have

∞

∞ √ √ ∗ |π(X)| = tdE(t) and |π(X) | = tdF (t), (2.6.12) 0

0

and by (2.6.9) that U, E(t), F (t) ∈ π(A) and F (t) = U E(t)U ∗

(2.6.13)

for all t ∈ [0, ∞). Since F (n)π(X) = |π(X)∗ |F (n)U ∈ B(H) for all n ∈ N, we can define an element Xn ∈ T ∗ (H) by Xn = (F (n)π(X), F (n)x, (E(n)y)∗ ),

n ∈ N.

48

2 Fundamentals of Observable Algebras

Since CA π(X) ⊂ π(X)CA and CA π(X)∗ ⊂ π(X)∗ CA by (2.6.8), we get that E(n)[λ(A)] ⊂ [λ(A)] and F (n)[λ(A)] ⊂ [λ(A)] for all n ∈ N. Hence it follows that ν(Xn ) = (F (n)x, (E(n)y)∗ ) ∈ H0 ,

(2.6.14)

and from (2.6.11) and (2.6.12) that π(Xn ) = |π(X)∗ |F (n)U ∈ π(A) , π(Xn )λ(A) = F (n)π(X)λ(A) = F (n)π(A)x = π(A)F (n)x, π(Xn )∗ λ(A ) = U ∗ |π(X)∗ |F (n)λ(A ) = U ∗ |π(X)∗ |U E(n)U ∗ λ(A ) = |π(X)|E(n)U ∗ λ(A ) = E(n)π(X)∗ λ(A ) = E(n)π(A)∗ y = π(A)∗ E(n)y for all A ∈ A. Therefore, Xn ∈ Aτ . Furthermore, we can prove that (λ(Xn )|λ(A )) = (λ(A)|λ(Xn ))

(2.6.15)

for all A ∈ A. Indeed, for α < 0 take a continuous bounded function f on [0, ∞): f (t) =

  1 1 t =− 1− , t −α α t −α

t ∈ [0, ∞).

Since f (|π(X)∗ |2 ) =



∞ 0 ∞

1 dF (t), t −α

t dF (t), t − α 0   ∞ √ t ∗ ∗ 2 ∗ dF (t) by (2.6.2), π(X) f (|π(X) | ) = U t −α 0

|π(X)∗ |2 f (|π(X)∗ |2 ) =

2.6 Commutants and Bicommutants of T ∗ -algebras

49

(see Theorem B.6 in Appendix B), they are bounded, and similarly f (|π(X)|2 ) and |π(X)|2 f (|π(X)|2 ) are both bounded. Hence it follows from (2.6.11) and (2.6.13) that 1 1 (f (|π(X)∗ |2 )x|λ(A )) = − (x|λ(A )) + (|π(X)∗ |2 f (|π(X)∗ |2 )x|λ(A )) α α 1 1 = − (λ(A)|y) + (π(X)∗ f (|π(X)∗ |2 )x|π(X)∗ λ(A )) α α 1 1 = − (λ(A)|y) + (π(X)∗ f (|π(X)∗ |2 )x|π(A)∗ y) α α 1 1 = − (λ(A)|y) + (π(X)∗ f (|π(X)∗ |2 )π(A)x|y) α α 1 1 = − (λ(A)|y) + (π(X)∗ f (|π(X)∗ |2 )π(X)λ(A)|y) α α 1 = − (λ(A)|y) α     ∞ 1 ∗ + tdF (t) U |π(X)|λ(A)|y |π(X)|U α 0 1 = − (λ(A)|y) α     ∞ 1 ∗ + tdU F (t)U |π(X)|λ(A)|y |π(X)| α 0 1 = − (λ(A)|y) α     ∞ 1 + tdE(t) |π(X)|λ(A)|y |π(X)| α 0 1 1 = − (λ(A)|y) + (|π(X)|f (|π(X)|2 )|π(X)|λ(A)|y) α α 1 = (λ(A)| − (I − |π(X)|2 )f (|π(X)|2 )y) α = (λ(A)|f (|π(X)|2 )y)

(2.6.16)

for all A ∈ A. Let χn be an indicator function of the closed interval [0, n] on [0, ∞). Since

n 1 f (t)dt = 1, t ∈ [0, n], χn (t) = log(−(n − α)|α) 0

50

2 Fundamentals of Observable Algebras

it follows from (2.6.16) that (λ(Xn )|λ(A )) = (F (n)x|λ(A )) = (χn (|π(X)∗ |2 )x|λ(A ))

n 1 = (f (|π(X)∗ |2 )x|λ(A ))dt log(−(n − α)|α) 0

n 1 = (λ(A)|f (|π(X)|2 )y)dt log(−(n − α)|α) 0 = (λ(A)|χn (|π(X)|2 )y) = (λ(A)|E(n)y) = (λ(A)|λ(Xn )) for all A ∈ A, which proves (2.6.15). Thus we get that Xn ∈ Aρ and ν(Xn ) ∈ H0 by (2.6.14), and lim ν(Xn ) = lim (F (n)x, (E(n)y)∗ )

n→∞

n→∞

= (x, y ∗ ) ∈ J ν(A)⊥ ∩ H0 . Thus, ν(Aρ ) ∩ H0 is dense in J ν(A)⊥ ∩ H0 . This completes the proof. The Proof of Proposition 2.6.5 By Lemma 2.6.6 we have ν(Aρρ ) ⊂ H0 .

(2.6.17)

Take an arbitrary ξ ∈ ν(Aρρ ). By Lemma 2.6.7 we can prove that ν(Aρρ ) ⊂ J ν(Aρ )⊥ ⊂ J (J ν(A)⊥ ∩ H0 )⊥ = (ν(A)⊥ ∩ H0 )⊥ , and by (2.6.17) that ξ ∈ H0 and ξ ∈ (ν(A)⊥ ∩ H0 )⊥ . Take an arbitrary η ∈ ν(A)⊥ . Then η is decomposed into η = η1 + η2 ,

η1 ∈ H0 , η2 ∈ H0⊥ .

(2.6.18)

2.6 Commutants and Bicommutants of T ∗ -algebras

51

By (2.6.17) we have η2 ∈ H0⊥ ⊂ ν(Aρρ )⊥ ⊂ ν(A)⊥ ; hence η1 = η − η2 ∈ ν(A)⊥ ∩ H0 . It therefore follows from (2.6.18) that (ξ |η) = (ξ |η1 ) + (ξ |η2 ) = 0, which implies that ξ ∈ ν(A)⊥⊥ = [ν(A)]. Thus ν(A) is dense in ν(Aρρ ). This completes the proof. Notes As for the basic definitions and the fundamental theorems (spectral resolution, functional calculus for a self-adjoint operator, polar decomposition of a densely defined closed operator in a Hilbert space), refer the reader to Appendix B.

Chapter 3

Density Theorems

In this chapter we will generalize the von Neumann density theorem and the Kaplansky density theorem (see Appendix C) which are the fundamental results of the theory of von Neumann algebras to the case of observable algebras. They play an important role in studies of observable algebras.

3.1 Von Neumann Type Density Theorem The purpose of this section is to prove the following von Neumann type density theorem. Theorem 3.1.1 (von Neumann Type Density Theorem) Let A be a T ∗ -algebra on H. Then, ¯ ∗ ] = AT := Aππ ∩ Aτ τ ∩ Aρρ . A[τ σs ¯ σ∗s ] coincides with the closure of A[τ ¯ u ] under the σ -strong∗ topology, Since A[τ we may assume without loss of generality that A is a CT ∗ -algebra. We shall prove this theorem after some preparations. Let H and K be Hilbert spaces. For A ∈ T ∗ (H) and B ∈ T ∗ (K) we define the tensor product A ⊗ B of A and B by   A ⊗ B = π(A) ⊗ π(B), λ(A) ⊗ λ(B), (λ(A ) ⊗ λ(B  ))∗ . (k)

Let l 2 = l 2 (Z+ ), Z+ = {0, 1, 2, . . .} and ek = (0, . . . , 0, 1 , 0, . . .), k ∈ Z+ . Put G = (I, e0 , e0∗ ) ∈ T ∗ (l 2 ).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Inoue, Tomita’s Lectures on Observable Algebras in Hilbert Space, Lecture Notes in Mathematics 2285, https://doi.org/10.1007/978-3-030-68893-6_3

53

54

3 Density Theorems

Then, we have, for A ∈ T ∗ (H), A ⊗ G = (π(A) ⊗ I, λ(A) ⊗ e0 , (λ(A ) ⊗ e0 )∗ ), and the map: ψ:

A → A ⊗ G

is a σ -strongly∗ homeomorphic ∗-isomorphism of T ∗ (H) into T ∗ (H ⊗ l 2 ). We ∞ now denote by HZ+ and HN the direct sums ⊕∞ k=0 Hk and ⊕k=1 Hk of {Hk }k∈Z+ and {Hk }k∈N , where Hk = H ⊗ ek , k ∈ Z+ , respectively. Then we  can regard an ∞ element g = (g ) of the Hilbert space H (resp. H ) as an element k Z N + k=0 gk ⊗ ek ∞ 2 (resp. k=0 gk ⊗ ek , e0 = 0) of the Hilbert space H ⊗ l . Then, for g = (gk ) ∈ HN and A ∈ T ∗ (H) we obtain ∞   Tg (A ⊗ G) = π(A) ⊗ I, λ(A) ⊗ e0 + π(A)gk ⊗ ek , (λ(A ) ⊗ e0 k=1

+

∞ 

 π(A)∗ gk ⊗ ek )∗ ,

k=1

so that ! λ(Tg (A ⊗ G)) =

∞ 

"1 2

π(A)gk 2 + λ(A)2

k=1

and ! ν(Tg (A ⊗ G)) =

∞ 

π(A)gk 2 +

k=1

∞ 

"1 π(A)∗ gk 2 + λ(A)2 + λ(A )2

2

k=1

for all A ∈ T ∗ (H), which implies the following Lemma 3.1.2 The σ -strong∗ topology τσ∗s on T ∗ (H) is the weakest topology for which the map: A ∈ T ∗ (H) → ν(Tg (A ⊗ G)) ∈ (H ⊗ l 2 ) ⊕ (H ⊗ l 2 )∗ is continuous for every g ∈ HN . For every element X of T ∗ (H ⊗ l 2 ), π(X) and λ(X) are represented as π(X) = (πj k (X))  πj k (X) ⊗ ej k ∈ B(H ⊗ l 2 ) = j,k∈Z+

3.1 Von Neumann Type Density Theorem

55

and λ(X) =



λk (X) ⊗ ek ∈ H ⊗ l 2 ,

k∈Z+

where πj k (X) ∈ B(H) and λk (X) ∈ H for j, k ∈ Z+ . For the commutants (A ⊗ G)π , (A ⊗ G)τ and (A ⊗ G)ρ we have the following Lemma 3.1.3 Let A be a CT ∗ -algebra on H. Then the following (1), (2) and (3) hold: (1) (A ⊗ G)π = (π(A) ⊗ B(l 2 ), H ⊗ l 2 , (H ⊗ l 2 )∗ ). (2) K ∈ (A ⊗ G)τ if and only if π(K) = (πj k (K))  = πj k (K) ⊗ ej k j,k∈Z+

and λ(K) =

∞ 

λk (K) ⊗ ek , λk (K) ∈ H

k=0

satisfy the following (i)–(iii): (i) πj k (K) ∈ π(A) for k, j ∈ Z+ . (ii) K00 := (π00 (K), λ0 (K), λ0 (K  )∗ ) ∈ Aτ . (iii) Take an arbitrary j ∈ N. We put Kj 0 = (πj 0 (K), λj (K), λj (K  )∗ ), K0j = (π0j (K), λj (K), λj (K  )∗ ) and

   |Kj 0 | = (Kj 0 Kj 0 )1/2 = |πj 0 (K)|, Uj∗ λj (K), (Uj∗ λj (K))∗ ,     |K0j | = (K0j K0j )1/2 = |π0j (K)∗ |, Vj∗ λj (K  ), (Vj∗ λj (K  ))∗ ,

where πj 0 (K) = Uj |πj 0 (K)| and π0j (K)∗ = Vj |π0j (K)∗ | are the polar decompositions of πj 0 (K) and π0j (K)∗ , respectively. Then, 

|Kj 0 |, |Kj 0 | ∈ Aτ

56

3 Density Theorems

and λj (K) = Uj λ(|Kj 0 |) + xj , 

λj (K  ) = Vj λ(|K0j |) + yj , where xj and yj are elements of H satisfying π(A)xj = 0 and π(A)yj = 0 for all A ∈ A. (3) (A ⊗ G)ρ = {K ∈ (A ⊗ G)τ ;

K00 ∈ Aρ }.

Proof (1) This is trivial. (2) Take an arbitrary K ∈ (A ⊗ G)τ . Then we have π(K)(π(A) ⊗ I ) = (π(A) ⊗ I )π(K),

(3.1.1)

π(K)(λ(A) ⊗ e0 ) = (π(A) ⊗ I )λ(K),

(3.1.2)

π(K)∗ (λ(A ) ⊗ e0 ) = (π(A)∗ ⊗ I )λ(K  )

(3.1.3)

for all A ∈ A. By (3.1.1) we get (i): πj k (K) ∈ π(A) for all j, k ∈ Z+ , and so (i) holds. By (3.1.2) and (3.1.3) we can prove that ∞ 

πj 0 (K)λ(A) ⊗ ej =

j =0 ∞ 

∞ 

π(A)λj (K) ⊗ ej ,

j =0

π0j (K)∗ λ(A) ⊗ ej =

j =0

∞ 

π(A)λj (K  ) ⊗ ej ,

j =0

so that πj 0 (K)λ(A) = π(A)λj (K),

(3.1.4)

π0j (K)∗ λ(A) = π(A)λj (K  )

(3.1.5)

for all j ∈ Z+ . In particular, we have π00 (K) ∈ π(A) , π00 (K)λ(A) = π(A)λ0 (K), π00 (K)∗ λ(A ) = π(A)∗ λ0 (K  ),

3.1 Von Neumann Type Density Theorem

57

which implies (ii): K00 = (π00 (K), λ0 (K), λ0 (K  )∗ ) ∈ Aτ . However, we cannot get Kj 0 ∈ Aτ from (3.1.4) and (3.1.5) because the equality: πj 0 (K)∗ λ(A ) = π(A)∗ λj (K  ) does not hold. Similarly, we cannot   get K0j ∈ Aτ . From this, we consider the absolutes |Kj 0 | and |K0j | of Kj 0 and K0j , and show that they are elements of Aτ . Since Uj , |πj 0 (K)| ∈ π(A) , it follows from (3.1.4) that 

|πj 0 (K)|λ(A) = Uj∗ πj 0 (K)λ(A) = Uj∗ π(A)λj (K) = π(A)Uj∗ λj (K) for all A ∈ A. Therefore   |Kj 0 | = |πj 0 (K)|, Uj∗ λj (K), (Uj∗ λj (K))∗ ∈ Aτ , and λj (K) = Uj λ(|Kj 0 |) + xj , where xj := (I − Uj Uj∗ )λj (K). Then, by (3.1.4) we obtain π(A)xj = (I − Uj Uj∗ )π(A)λj (K) = (I − Uj Uj∗ )πj 0 (K)λ(A) =0 

for all A ∈ A. Furthermore, we can prove in the same way as Kj 0 that |K0j | =    |π0j (K)∗ |, Vj∗ λj (K  ), (Vj∗ λj (K  ))∗ ∈ Aτ and λj (K  ) = Vj λ(|K0j |) + yj , where yj := (I − Vj Vj∗ )λj (K  ). Then π(A)yj = 0 for all A ∈ A. Thus (iii) holds. Suppose, conversely, that K ∈ T ∗ (H ⊗ l 2 ) satisfies statements (i)–(iii). Then, since π(A)xj = 0 and π(A)yj = 0 for all A ∈ A and j ∈ N, we have (π(A) ⊗ I )λ(K) = (π(A) ⊗ I )(

∞ 

λj (K) ⊗ ej )

j =0

= π(A)λ0 (K) ⊗ e0 +

∞ 

π(A)λj (K) ⊗ ej

j =1

= π00 (K)λ(A) ⊗ e0 +

∞  j =1

π(A)(Uj λ(|Kj 0 |) + xj ) ⊗ ej

58

3 Density Theorems

= π00 (K)λ(A) ⊗ e0 +

∞ 

π(A)Uj λ(|Kj 0 |) ⊗ ej

j =1

= π00 (K)λ(A) ⊗ e0 +

∞ 

Uj π(|Kj 0 |)λ(A) ⊗ ej

j =1

= π00 (K)λ(A) ⊗ e0 +

∞ 

πj 0 (K)λ(A) ⊗ ej

j =1

=

∞ 

πj 0 (K)λ(A) ⊗ ej

j =0

= π(K)(λ(A) ⊗ e0 ) and ∗

(π(A) ⊗ I )λ(K ) = 

∞ 

π(A)∗ λj (K  ) ⊗ ej

j =0

= π(A)∗ λ0 (K  ) +

∞ 

π(A)∗ (Vj λ(|K0j |) + yj ) ⊗ ej 

j =1

= π00 (K)∗ λ(A ) ⊗ e0 +

∞ 



Vj π(|K0j |)λ(A ) ⊗ ej

j =1

=

∞ 

π0j (K)∗ λ(A ) ⊗ ej

j =0

= π(K)∗ (λ(A ) ⊗ e0 ). Therefore, K ∈ (A ⊗ G)τ . (3) This follows from the following: K ∈ (A ⊗ G)ρ

iff K ∈ (A ⊗ G)τ and (λ(A ⊗ G)|λ(K  ))=(λ(K)|λ((A ⊗ G) )) for all A ∈ A iff K ∈ (A ⊗ G)τ 

and (λ(A)|λ(K00 )) = (λ(K00 )|λ(A )) for all A ∈ A iff K ∈ (A ⊗ G)τ and K00 ∈ Aρ .

This completes the proof.

3.1 Von Neumann Type Density Theorem

59

Lemma 3.1.4 Let A be a CT ∗ -algebra on H and AT = Aππ ∩ Aτ τ ∩ Aρρ . Then AT ⊗ G = (A ⊗ G)T . Proof By the equality:   (A ⊗ G)ππ = π(A) ⊗ I, H ⊗ l 2 , (H ⊗ l 2 )∗ ,

(3.1.6)

we have AT ⊗ G ⊂ (A ⊗ G)ππ .

(3.1.7)

Suppose A ∈ AT and K ∈ (A ⊗ G)τ . Then, since A ∈ Aτ τ and π(A)xj = 0 for all j ∈ Z+ , where xj is as in Lemma 3.1.3, and PA = PAτ τ by Theorem 4.1.7 in Sect. 4.1, it follows that PAτ τ xj = PA xj = 0, which implies that π(A)xj = 0 for all j ∈ Z+ . Hence we can show in the same as the proof of Lemma 3.1.3, (2) that (π(A) ⊗ I )λ(K) = π(K)λ(A ⊗ G) and (π(A) ⊗ I )∗ λ(K  ) = π(K)∗ λ((A ⊗ G) ). Therefore A ⊗ G ∈ (A ⊗ G)τ τ . Thus AT ⊗ G ⊂ (A ⊗ G)τ τ .

(3.1.8)

Furthermore, for any A ∈ AT and K ∈ (A ⊗ G)ρ , it follow from Lemma 3.1.3 that (λ(A ⊗ G)|λ(K  )) = (λ(A) ⊗ e0 |λ(K  )) 

= (λ(A)|λ(K0 )) = (λ(K0 )|λ(A )) = (λ(K)|λ((A ⊗ G) )). Hence, A ⊗ G ∈ (A ⊗ G)ρρ , so AT ⊗ G ⊂ (A ⊗ G)ρρ . By (3.1.7)–(3.1.9) we have AT ⊗ G ⊂ (A ⊗ G)T .

(3.1.9)

60

3 Density Theorems

Conversely, take an arbitrary X ∈ (A ⊗ G)T . By (3.1.6) there exists an element X0 of π(A) such that π(X) = X0 ⊗ I . Furthermore, since XK = KX for all K ∈ (A ⊗ G)τ , that is, (X0 ⊗ I )λ(K) = π(K)λ(X), (X0∗ ⊗ I )λ(K  ) = π(K)∗ λ(X ), we have X0 λj (K) =

∞ 

πj k (K)λk (X),

(3.1.10)

πj k (K)∗ λk (X ).

(3.1.11)

k=0

X0 λj (K  ) =

∞  k=0 

For any K00 = (π(K00 ), λ(K00 ), λ(K00 )) ∈ Aτ and j ∈ N, we define an element K = (π(K), λ(K), λ(K  )∗ ) of T ∗ (H ⊗ l 2 ) as follows: π(K) = π(K00 ) ⊗ e00 + I ⊗ e1j , (j > 0), λ(K) = λ(K00 ) ⊗ e0 , 

λ(K  ) = λ(K00 ) ⊗ e0 . Then, by Lemma 3.1.3, (2) we get K ∈ (A ⊗ G)τ . Take an arbitrary j > 1. Since X0 λ(K00 ) ⊗ e0 = (X0 ⊗ I )λ(K) = π(K)λ(X) = (π(K00 ) ⊗ e00 + I ⊗ e1j )(



λk (X) ⊗ ek )

k

= π(K00 )λ0 (X) ⊗ e0 + λj (X) ⊗ e1 , it follows that X0 λ(K00 ) = π(K00 )λ0 (X) and λj (X) = 0.

(3.1.12)

Similarly, we see that X0∗ λ(K00 ) = π(K00 )∗ λ0 (X ) and λj (X ) = 0. 

(3.1.13)

3.1 Von Neumann Type Density Theorem

61

Therefore (X0 , λ0 (X), λ0 (X )∗ ) ∈ Aτ τ .

(3.1.14)

Furthermore, it follows from (3.1.12) and (3.1.13) that π(X) = X0 ⊗ I, λ(X) =

∞ 

λj (X) ⊗ ej = λ0 (X) ⊗ e0 ,

j =0

λ(X ) =

∞ 

λj (X ) ⊗ ej = λ0 (X ) ⊗ e0 ,

j =0

which implies by (3.1.14) that X ∈ Aτ τ ⊗ G. Furthermore, we get (λ(X)|λ(K  )) = (λ(K)|λ(X )) for all K ∈ (A ⊗ G)ρ , which implies since Aρ ⊗ G ⊂ (A ⊗ G)ρ that 

(λ0 (X)|λ(K0 )) = (λ(K0 )|λ0 (X )) for all K0 ∈ Aρ . Hence, (X0 , λ0 (X), λ0 (X )∗ ) ∈ Aρρ , so that X ∈ Aρρ ⊗ G and X ∈ AT ⊗ G. Thus we have (A ⊗ G)T ⊂ AT ⊗ G. This completes the proof. The Proof of Theorem 3.1.1 By Proposition 2.6.3 and Lemma 3.1.4 we have Tg (A ⊗ G) ⊂ Tg (AT ⊗ G) = (Tg (A ⊗ G))T ⊂ Tg (A ⊗ G)ρρ

(3.1.15)

for any g ∈ G. Since ν(Tg (A ⊗ G)) is dense in ν((Tg (A ⊗ G))ρρ ) by Proposition 2.6.5, it follows from (3.1.15) that ν(Tg (A ⊗ G)) is dense in ν(Tg (AT ⊗ G)), which implies by Lemma 3.1.2 that A is strongly∗ dense in AT . Furthermore, since ¯ σ∗s ] = AT . This completes the proof. AT is strongly∗ closed in T ∗ (H), we have A[τ Theorem 3.1.1 proves the following Corollary 3.1.5 Let A be a T ∗ -algebra on H. Then AT coincides with each of the closures of A under the topologies τw , τσ w , τs , τσ s , τs∗ and τσ∗s .

62

3 Density Theorems

Proof By Proposition 2.6.2, AT is weakly closed, and furthermore by (2.5.1) we have ¯ σ∗s ] ⊂ A[τ ¯ ] ⊂ A[τ ¯ w ] ⊂ AT , A[τ where τ is one of the topologies τσ w , τs , τσ s and τs∗ , which implies by Theorem 3.1.1 that ¯ ] = A[τ ¯ w ] = AT . ¯ ∗ ] = A[τ A[τ σs This completes the proof. Definition 3.1.6 A T ∗ -algebra A on H is called a W T ∗ -algebra if A = AT , ¯ ], where τ is any of the topologies τw , τσ w , τs , τσ s , τ ∗ and equivalently, A = A[τ s ∗ τσ s . It is clear that if A is a W T ∗ -algebra, then it is a CT ∗ -algebra.

3.2 Kaplansky Type Density Theorem In this section we prove the Kaplansky type density theorem using the functional # calculus theorem in Sect. 2.3. For x = (xn ) ∈ HN := ∞ H n=1 n (where Hn = H, n ∈ N), we define a seminorm px on T ∗ (H) by  px (A)=

∞  n=1

∞  π(A)xk  + π(A)∗ xk 2 +λ(A)2+λ(A )2 2

1 2

, A ∈ T ∗ (H).

n=1

As stated in Sect. 2.5, the σ -strong∗ topology τσ∗s on T ∗ (H) is defined by a family {px ; x ∈ HN } of seminorms, and T ∗ (H)[τσ∗s ] is a complete locally convex ∗¯ σ∗s ] the σ -strong∗ closure of A, and algebra. For A ⊂ T ∗ (H) we denote by A[τ 0 denote by u(A) and u(A) the unit balls of A, that is, u(A) := {A ∈ A; A  1}, u(A)0 := {A ∈ A; A < 1}. The purpose of this section is to prove the following result called Kaplansky type density theorem. Theorem 3.2.1 (Kaplansky Type Density Theorem) Let A be a T ∗ -algebra on H. Then ¯ ∗ ]). u(A)[τσ∗s ] = u(A[τ σs

3.2 Kaplansky Type Density Theorem

63

For the proof of this theorem we may assume without loss of generality that A is a ¯ u ]). We first verify the following CT ∗ -algebra because u(A)[τu ] = u(A[τ Lemma 3.2.2 Put uπ (A) := {A ∈ A; π(A)  1}. Then ¯ ∗ ]). uπ (A)[τσ∗s ] = uπ (A[τ σs Proof We consider continuous functions f and g defined by 2 , t  0, 1+t 1 g(t) = √ , 0  t  1. 1+ 1−t

f (t) =

Then it follows from Theorem 2.3.11 that for any A ∈ T ∗ (H)   F (A) := f (AA )A = π(F (A)), λ(F (A)), λ(F (A) )∗ ∈ uπ (CT ∗ (A)),

(3.2.1)

where π(F (A)) = 2(I + π(A)π(A)∗)−1 π(A), λ(F (A)) = 2(I + π(A)π(A)∗)−1 λ(A), λ(F (A) ) = 2(I + π(A)∗ π(A))−1 λ(A ), and that for any A ∈ uπ (T ∗ (H)),   G(A) := g(AA )A = π(G(A)), λ(G(A)), λ(G(A) )∗ ∈ uπ (CT ∗ (A)), where   1 −1 π(G(A)) = I + (I − π(A)π(A)∗ ) 2 π(A),   1 −1 λ(A) λ(G(A)) = I + (I − π(A)π(A)∗ ) 2   1 −1 λ(A ). λ(G(A) ) = I + (I − π(A)∗ π(A)) 2

(3.2.2)

64

3 Density Theorems

Let A ∈ uπ (T ∗ (H)). Since 

−1 I + π(G(A))π(G(A))∗   −1  −1 −1 ∗ 12 ∗ ∗ 12 = I + I +(I − π(A)π(A) ) π(A)π(A) I +(I − π(A)π(A) ) =

 1 1 I + (I − π(A)π(A)∗ ) 2 , 2

it follows that −1  π(G(A)) π(F (G(A))) = 2 I + π(G(A))π(G(A))∗     1 1 −1 = I + (I − π(A)π(A)∗) 2 I + (I − π(A)π(A)∗ ) 2 π(A) = π(A), and that  −1 λ(F (G(A))) = 2 I + π(G(A))π(G(A))∗ λ(G(A))     1 1 −1 = I + (I − π(A)π(A)∗) 2 I + (I − π(A)π(A)∗ ) 2 λ(A) = λ(A). Furthermore, since 

 −1   −1 1 −2 ∗ ∗ 2 = I + π(A) I + (I − π(A) π(A)) π(A) I + π(G(A)) π(G(A)) ∗



  −1 1 −2 ∗ = I + π(A) π(A) I + (I − π(A) π(A)) 2 =

∗

 1 1 I + (I − π(A)∗ π(A)) 2 , 2

it follows that    1 1 −1 λ(F (G(A) )) = I + (I − π(A)∗ π(A)) 2 I + (I − π(A)∗ π(A)) 2 λ(A ) = λ(A ). Thus we have F (G(A)) = A.

(3.2.3)

3.2 Kaplansky Type Density Theorem

65

By (3.2.1)–(3.2.3) we have F (B) = uπ (B) for every CT ∗ -algebra B on H. Since A is a CT ∗ -algebra by assumption and ¯ ∗ ]) is a CT ∗ -algebra too, we have A([τ σs F (A) = uπ (A)

(3.2.4)

¯ σ∗s ]). ¯ σ∗s ]) = uπ (A[τ F (A[τ

(3.2.5)

and

Furthermore, the map F :

A ∈ T ∗ (H) −→ F (A) ∈ uπ (T ∗ (H))

(3.2.6)

is τσ∗s -continuous. Indeed, for any A, B ∈ T ∗ (H) we have 1 (π(F (A)) − π(F (B))) 2   −1 −1 = I + π(A)π(A)∗ π(A) − I + π(B)π(B)∗ π(B)   −1 −1 = I + π(A)π(A)∗ π(A) − π(B) I + π(B)∗ π(B)  −1 = I + π(A)π(A)∗ % −1 $ , (π(A) − π(B)) + π(A)(π(B)∗ − π(A)∗ )π(B) I + π(B)∗ π(B) which yields that for any x = (xn ) ∈ HN ∞ 

π(F (A))xn − π(F (B))xn 2

n=1

∞   −1  −1 4  I + π(A)π(A)∗ (π(A) − π(B)) I + π(B)∗ π(B) xn n=1

 −1   −1 + I+π(A)π(A)∗ π(A)(π(B)∗−π(A)∗)π(B) I+π(B)∗ π(B) xn 2 ∞  ∞   2 ∗ ∗ 2 =4 π(A)un − π(B)un  + π(B) vn − π(A) vn  , n=1

n=1

66

3 Density Theorems

where un := (I + π(B)∗ π(B))−1 xn ,  −1 vn := π(B) I + π(B)∗ π(B) xn and {un }, {vn } ∈ HN . Furthermore, we can verify that 1 λ(F (A)) − λ(F (B)) 2 = (I + π(A)π(A)∗)−1 λ(A) − (I + π(B)π(B)∗ )−1 λ(B) = (I + π(A)π(A)∗)−1 (λ(A) − λ(B))   + (I + π(A)π(A)∗ )−1 − (I + π(B)π(B)∗ )−1 λ(B)  λ(A) − λ(B) + (I + π(A)π(A)∗ )−1 (π(B)π(B)∗ −π(A)π(A)∗ )(I + π(B)π(B)∗ )−1 λ(B)  λ(A) − λ(B) + (I + π(A)π(A∗ ))−1 π(A)(π(B)∗ −π(A)∗ )(I + π(B)π(B)∗ )−1 λ(B) +(I + π(A)π(A)∗)−1 (π(B) − π(A))π(B)∗ (I + π(B)π(B)∗ )−1 λ(B)  λ(A) − λ(B) + (π(A)∗ − π(B)∗ )(I + π(B)π(B)∗ )−1 λ(B) +(π(A) − π(B))π(B)∗ (I + π(B)π(B)∗ )−1 λ(B), and similarly, 1 λ(F (A) ) − λ(F (B  ))  λ(A ) − λ(B  ) 2 +(π(A) − π(B))(I + π(B)∗ π(B))−1 λ(B  ) +(π(B)∗ − π(A)∗ )π(B)(I + π(B)∗ π(B))−1 λ(B  ).

Hence F is τσ∗s -continuous. Using these facts, we can prove that ¯ ∗ ]). uπ (A)[τσ∗s ] = uπ (A[τ σs ¯ σ∗s ]) is τσ∗s -closed. Hence, Indeed, it is clear that uπ (A[τ ¯ σ∗s ]). uπ (A)[τσ∗s ] ⊂ uπ (A[τ

(3.2.7)

3.2 Kaplansky Type Density Theorem

67

¯ ∗ ]). By (3.2.5), A = F (B) for some Conversely, take an arbitrary A ∈ uπ (A[τ σs ∗ ∗ ¯ B ∈ A[τ σ s ]. Then we can take a net {Bα } in A such that τσ s -limα Bα = B, so that by (3.2.6) τσ∗s − lim F (Bα ) = F (B) = A. α

Since F (Bα ) ∈ uπ (A) by (3.2.4), we have A ∈ uπ (A)[τσ∗s ]. Thus (3.2.7) holds. This complete the proof. The Proof of Theorem 3.2.1 Put uπ (A)0 = {A ∈ A; π(A) < 1}. Take an arbitrary A ∈ uπ (A). Since εA = (επ(A), ελ(A), ελ(A )∗ ) ∈ uπ (A)0 for each ε ∈ (0, 1) and τσ∗s - lim εA = A, we have A ∈ uπ (A)0 [τσ∗s ]. Therefore ε→0

uπ (A)0 [τσ∗s ] = uπ (A)[τσ∗s ].

(3.2.8)

Similarly, we can verify that u(A)0 [τσ∗s ] = u(A)[τσ∗s ].

(3.2.9)

¯ σ∗s ])0 . Then, since Take an arbitrary A ∈ u(A[τ ¯ σ∗s ]) = uπ (A)[τσ∗s ] = uπ (A)0 [τσ∗s ] A ∈ uπ (A[τ by Lemma 3.2.2 and (3.2.8), it follows that for any x = (xn ) ∈ HN and 0 < ε < min(1 − λ(A), 1 − λ(A )) there exists an element B of uπ (A)0 such that ∞ ∞   2 px (A − B) = π(A)xn − π(B)xn  + π(A)∗ xn − π(B)∗ xn 2 n=1

n=1

1 +λ(A) − λ(B)2 + λ(A ) − λ(B  )2 < ε. Hence we get π(B) < 1, λ(B) < λ(A) + ε < 1, λ(B  ) < λ(A ) + ε < 1,

2

68

3 Density Theorems

so that B ∈ u(A)0 and A ∈ u(A)0 [τσ∗s ]. Thus we have ¯ σ∗s ]) ⊂ u(A)0 [τσ∗s ], u(A[τ which yields by (3.2.9) that ¯ σ∗s ]) = u(A[τ ¯ σ∗s ])0 [τσ∗s ] ⊂ u(A)0 [τσ∗s ] = u(A)[τσ∗s ]. u(A[τ The converse inclusion is trivial. This completes the proof. Corollary 3.2.3 Let A be a T ∗ -algebra on H. Then ¯ ]) = u(A[τ ¯ ∗ ]) = u(A)[τ ∗ ], u(A)[τ ] = u(A[τ σs σs where τ is any of the topologies τw , τσ w , τs , τσ s and τs∗ . In particular, if A is a W T ∗ -algebra, then the unit ball u(A) is closed under these topologies. ¯ w ]) is weakly closed. Hence it follows from Proof It is easily shown that u(A[τ Corollary 3.1.5 that ¯ w ]) = u(A[τ ¯ ∗ ]), u(A)[τσ∗s ] ⊂ u(A)[τ ] ⊂ u(A)[τw ] ⊂ u(A[τ σs which completes the proof. For a closed ∗-ideal N(A) of a CT ∗ -algebra A defined by N(A)= {A ∈ A; π(A) = 0}, we have the following Corollary 3.2.4 Let A be a CT ∗ -algebra on H. Then ¯ ∗ ]) = N(A). N(A[τ σs ¯ σ∗s ]). By Lemma 3.2.2, Proof Take an arbitrary A ∈ N(A[τ ¯ σ∗s ]) := {A ∈ A[τ ¯ σ∗s ]; π(A)  ε} A ∈ uεπ (A[τ = uεπ (A)[τσ∗s ] for any ε > 0. Hence, for any x ∈ HN there exists an element B of uεπ (A) such that px (A − B) < ε, which implies that π(B)  ε, λ(B) − λ(A) < ε, λ(B  ) − λ(A ) < ε, ¯ u ] = A, which completes the proof. that is, A − B  ε. Hence, A ∈ A[τ By Corollary 3.2.4 and Theorem 3.2.1, if A is a W T ∗ -algebra (see Definition 3.1.6), then N(A) is closed under all of topologies τw , τσ w , τs , τσ s , τs∗ and τσ∗s .

Chapter 4

Structure of CT ∗ -Algebras

Chapter 4 is devoted to structure of CT ∗ -algebras. In Sect. 4.1 it is proved that every CT ∗ -algebra A is decomposed into A = P (A) + Z(A) by the nondegenerate part P (A) and the nulifier Z(A) of A using the corresponding projections PA and NA , and more decompositions of A are obtained decomposing the projections PA and NA . Furthermore, the important equalities: PA = I − NAτ and PAτ = I − NA are verified. In Sect. 4.2 the notions of regular, semifinite, nondegenerate, singular and nilpotent CT ∗ -algebras are defined and their characterizations are investigated. In Sect. 4.3 it is shown that every commutative semisimple CT ∗ -algebra is isomorphic to the Banach ∗-algebra C0 () ∩ L2 (, μ) consisting of the commutative C ∗ algebra C0 () of continuous functions on a locally compact space  vanishing at infinity on  and the L2 -space L2 (, μ) defined by a regular Borel measure μ on . The following subjects are discussed: Sect. 4.4.1 definition of regular, (semisimple, ˜ nondegenerate, singular and nilpotent) T ∗ -algebras. Sect. 4.4.2 construction of λ(A) ∗ from λ(A). Sect. 4.4.3 construction of semisimple CT -algebras from the C ∗ algebra obtained by the uniform closure of π(A). Sect. 4.4.4 the semisimplicity and the singularity of T ∗ -algebras. 3.4.6 Construction of a natural weight on the positive cone (π(A) )+ of the von Neumann algebra π(A) . Sect. 4.4.6 the density of a ∗-subalgebra of a T ∗ -algebra under the strong∗ topology and under the uniform topology.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Inoue, Tomita’s Lectures on Observable Algebras in Hilbert Space, Lecture Notes in Mathematics 2285, https://doi.org/10.1007/978-3-030-68893-6_4

69

4 Structure of CT ∗ -Algebras

70

4.1 Decomposition of CT ∗ -Algebras Let A be a CT ∗ -algebra on H. Write P (A) =

the closed linear span of {AB; A, B ∈ A},

N(A) = {A ∈ A; π(A) = 0}, Z(A) = {A ∈ A; AX = XA = O for all X ∈ A}. Then, it is simple to show the following Proposition 4.1.1 Suppose that A is a CT ∗ -algebra on H. Then P (A), N(A) and Z(A) are closed ∗-ideals of A and N(A) is identical with the radical R(A) of A, which contains Z(A), where R(A) := {A ∈ A; (αA + XA) has an inverse in the unitization A11 of A for all α ∈ C and X ∈ A}. Remark 4.1.2 The map: A + N(A) ∈ A/N(A) → π(A) ∈ B(H) is a continuous ∗-isomorphism of the quotient Banach ∗-algebra A/N(A) onto the C ∗ -normed algebra π(A). For the definition of C ∗ -normed algebra see C.2 in Appendix. If π(A) is a C ∗ -algebra, then A/N(A) is ∗-isomorphic to π(A) as Banach ∗-algebras, so that A/N(A) is a C ∗ -algebra. In order to prove that every CT ∗ -algebra A on H is decomposed into P (A) + Z(A), we define the following projections PA and NA by PA = Proj [π(A)H], NA = Proj [λ(N(A))]. Then, since [π(A)H] is both π(A)- and π(A) -invariant, and [λ(N(A))] is π(A)invariant, we have PA ∈ π(A) ∩ π(A) and NA ∈ π(A) . Furthermore, we get, by Corollary C.3 in Appendix C, the following result used often in this note. There exists a net {eα } in uπ (A) such that π(eα ) converges strongly to PA .

(4.1.1) For any projection E in π(A) we put τE (A) = (π(A), Eλ(A), (Eλ(A ))∗ ), νE (A) = (0, Eλ(A), (Eλ(A ))∗ )

4.1 Decomposition of CT ∗ -Algebras

71

for A ∈ A, and τE (A) = {τE (A); A ∈ A}, νE (A) = {νE (A); A ∈ A}. Then, the following is immediate. Lemma 4.1.3 Let A be a CT ∗ -algebra on H. Then (1) τE is a continuous ∗-homomorphism of the Banach ∗-algebra A into the Banach ∗-algebra T ∗ (H), and τE (A) is a T ∗ -algebra on H. If τE (A) ⊂ A, then τE (A) is a CT ∗ -algebra on H; (2) νE (A) is a T ∗ -algebra on H satisfying νE (A)2 = {O}. If νE (A) ⊂ A, then νE (A) is a CT ∗ -algebra on H. Under these preparations we get a decomposition theorem of a CT ∗ -algebra. Theorem 4.1.4 Let A be a CT ∗ -algebra on H. Then (1) P (A) = τPA (A); (2) Z(A) = νI −PA (A); (3) A = P (A) + Z(A). Proof (1) Let A ∈ A and denote by PA the projection onto the subspace [π(A)H]. By Theorem 2.3.12, A is decomposed into As + An , where As = (π(A), PA λ(A), (PA λ(A ))∗ ) ∈ P (CT ∗ (A)), An = (0, (I − PA )λ(A), ((I − PA )λ(A ))∗ ) ∈ N(CT ∗ (A)). Since PA  PA , we have τPA (As ) = (π(A), PA λ(A), (PA λ(A ))∗ ) = As ∈ P (A) and τPA (An ) = (0, PA λ(An ), (PA λ(An ))∗ ). Furthermore, since eα An + An eα = (0, π(eα )(I − PA )λ(An ), (π(eα )(I − PA )λ(An ))∗ ) ∈ P (A)

4 Structure of CT ∗ -Algebras

72

and by (4.1.1) τu - lim(eα An + An eα ) = (0, PA λ(An ), (PA λ(An ))∗ ) α

= τPA (An ), where {eα } is a net in uπ (A) in (4.1.1), it follows that τPA (An ) ∈ P (A), which implies that τPA (A) = As + τPA (An ) ∈ P (A). Thus τPA (A) ⊂ P (A). Hence it follows from Lemma 4.1.3 that τPA (A) is a CT ∗ -algebra on H, so that the converse inclusion P (A) ⊂ τPA (A) is trivial. (2) Take an arbitrary A ∈ Z(A). Then, since π(A) = 0, π(eα )λ(A) = 0 and π(eα )λ(A ) = 0 for all α, it follows from (4.1.1) that PA λ(A) = 0 and PA λ(A ) = 0; hence A = νI −PA (A). Thus Z(A) ⊂ νI −PA (A). We show the converse. By (1) we see that νI −PA (A) = A − τPA (A) ∈ A for all A ∈ A, and that BνI −PA (A) = (0, π(B)(I − PA )λ(A), 0) = (0, (I − PA )π(B)λ(A), 0) = O, and that νI −PA (A)B = (0, 0, (π(B)∗ (I − PA )λ(A ))∗ ) =O for all A, B ∈ A, which implies that νI −PA (A) ⊂ Z(A). Thus, (2) is proved. (3) By (1) and(2) we obtain A = τPA (A) + νI −PA (A) ∈ P (A) + Z(A) for all A ∈ A. Hence, A ⊂ P (A) + Z(A). The converse is trivial. Thus, A = P (A) + Z(A). This completes the proof.

4.1 Decomposition of CT ∗ -Algebras

73

To consider more detailed decompositions of A, we define the following projections: FA = PA (I − NA ), SA = PA NA , ZA = (I − PA )NA , VA = (I − PA )(I − NA ). Then, FA , SA , ZA and VA are projections in π(A) , and I = PA + (I − PA ) = (FA + SA ) + (ZA + VA ) = FA + NA + VA . Lemma 4.1.5 (1) τFA (A) is a closed ∗-subalgebra of the CT ∗ -algebra P (A)[τu ], and τFA is a continuous ∗-homomorphism of the CT ∗ -algebra A[τu ] onto the CT ∗ -algebra τFA (A)[τu ] with kernel N(A). Hence, τFA (A) is ∗-isomorphic to A/N(A) . (2) νSA (A) = N(SA ) := {(0, x, y ∗ ); x, y ∈ SA H} = P (A) ∩ N(A). (3) νVA (A) = {O}. Proof (2) Take arbitrary x, y ∈ SA H. Since NA x = x and NA y = y, we can find sequences {An } and {Bn } in N(A) such that lim λ(An ) = x and lim λ(Bn ) = y. n→∞

n→∞

Then, we get that eα An + Bn eα = (0, π(eα )λ(An ), (π(eα )λ(Bn ))∗ ) ∈ P (A) ∩ N(A) for all α and n ∈ N, where {eα } is a net in A in (4.1.1) and that τu − lim lim (eα An + Bn eα ) = (0, PA x, (PA y)∗ ) α n→∞

= (0, x, y ∗ ). Hence, A := (0, x, y ∗ ) ∈ P (A) ∩ N(A) and νSA (A) = A ∈ νSA (A). Thus we have N(SA ) = νSA (A) and N(SA ) ⊂ P (A) ∩ N(A).

(4.1.2)

4 Structure of CT ∗ -Algebras

74

Conversely, suppose A ∈ P (A) ∩ N(A). Then, since A = (0, NA λ(A), (NA λ(A ))∗ ) ∈ P (A), it follows that A = τPA (A) = (0, PA NA λ(A), (PA NA λ(A ))∗ ) = (0, SA λ(A), (SA λ(A ))∗ ) = νSA (A) ∈ N(SA ), which implies by (4.1.2) that N(SA ) = P (A) ∩ N(A). (1) By Theorem 4.1.4 and (2) we see τFA (A) = τPA (A) + νSA (A) ∈ P (A) for all A ∈ A, which yields by Lemma 4.1.3 that τFA (A) is a closed ∗-subalgebra of the CT ∗ -algebra P (A)[τu ], and τFA is a continuous ∗-homomorphism of the CT ∗ algebra A[τu ] onto the CT ∗ -algebra τFA (A)[τu ]. It is easily shown that the kernel of τFA coincides with N(A). Hence τFA (A) is isomorphic to A/N(A). (3) By Theorem 2.3.12, A is decomposed into A = As + An , where As ∈ P (A) and An ∈ N(A). Therefore VA λ(A) = (I − PA )(I − NA )λ(As ) + (I − PA )(I − NA )λ(An ) = 0. This completes the proof. By Theorem 4.1.4 and Lemma 4.1.5 we obtain the following decomposition theorem of a CT ∗ -algebra which is one of the main results of this note. Theorem 4.1.6 Let A be a CT ∗ -algebra on H. Then P (A) = τFA (A) + νSA (A), Z(A) = νI −PA (A) = νZA (A), N(A) = νSA (A) + νZA (A) = νSA (A) + Z(A), and A = P (A) + Z(A) = (τFA (A) + νSA (A)) + Z(A) = τFA (A) + N(A).

4.1 Decomposition of CT ∗ -Algebras

75

For the relationship among the projections PA , NA , PAτ , NAτ , PAτ τ and NAτ τ we obtain the following important results: Theorem 4.1.7 Let A be a CT ∗ -algebra on H. Then (1) PAτ τ = PA and NAτ τ = NA ; (2) PA = I − NAτ and PAτ = I − NA . To prove Theorem 4.1.7 we prepare some statements. Lemma 4.1.8 Suppose K ∈ N(T ∗ (H)). Then, K ∈ Aτ if and only if PA λ(K) = PA λ(K  ) = 0. Proof It is easily shown that K ∈ Aτ if and only if π(A)λ(K) = π(A)λ(K  ) = 0 for all A ∈ A. (4.1.3) Suppose that K ∈ Aτ . Then it follows from (4.1.1) and (4.1.3) that PA λ(K) = lim π(eα )λ(K) = 0, α

PA λ(K ) = lim π(eα )λ(K  ) = 0. 

α

Conversely, suppose PA λ(K) = PA λ(K  ) = 0. Then it follows that π(A)λ(K) = PA π(A)λ(K) = π(A)PA λ(K) = 0, π(A)λ(K  ) = π(A)PA λ(K  ) = 0 for all A ∈ A, which implies by (4.1.3) that K ∈ Aτ . This completes the proof. Lemma 4.1.9 NAτ = I − PA . Proof By Lemma 4.1.8 we have (I − PA )λ(K) = λ(K) = NAτ λ(K) for all K ∈ N(Aτ ). Hence it follows that I − PA  NAτ . Conversely, take an arbitrary x ∈ H such that (I − PA )x = x. Then, by Lemma 4.1.8, x˜ := (0, x, 0) ∈ N(Aτ ). Hence, NAτ x = x, so I − PA  NAτ . This completes the proof. Lemma 4.1.10 PAτ τ = PA . Proof By Lemma 4.1.9, we get I − PAτ τ = NAτ τ τ = NAτ = I − PA , which completes the proof.

4 Structure of CT ∗ -Algebras

76

Lemma 4.1.11 SAτ τ = SA . Proof Let x ∈ SAτ τ H and write x˜ = (0, x, 0). By Lemma 4.1.10 we get PA x = x.

(4.1.4)

It is clear that x˜ ∈ Aππ . By Lemma 4.1.5, x˜ ∈ N(SAτ τ ) = P (Aτ τ ) ∩ N(Aτ τ ) ⊂ Aτ τ . Hence, K x˜ = xK ˜ for all K ∈ Aρ (⊂ Aτ ). Furthermore, for any K ∈ Aρ we can show that (λ(K)|λ(x˜  )) = 0, and that by (4.1.1),  ˜ )|λ(eα )) lim(π(x)λ(K α

= lim(x|π(K)∗ λ(eα )) α

= lim(π(eα )x|λ(K  )) α

= (x|λ(K  ))  = (λ(x)|λ(K ˜ ))

= 0. Hence, x˜ ∈ Aρρ . Thus x˜ ∈ Aππ ∩ Aτ τ ∩ Aρρ = AT . This implies by the von Neumann type density theorem (Theorem 3.1.1) and Corollary 3.2.4 that ¯ ∗ ]) = N(A). x˜ ∈ N(AT ) = N(A[τ σs Hence it follows from (4.1.4) that SA x = x. Therefore SAτ τ = SA . The Proof of Theorem 4.1.7 By Lemma 4.1.10 we get that PAτ τ = PA ,

(4.1.5)

and that by Lemmas 4.1.10 and 4.1.11, PA (NAτ τ − NA ) = PAτ τ NAτ τ − PA NA = SAτ τ − SA = 0.

(4.1.6)

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77

By (4.1.3), V˜A := (VA , 0, 0) ∈ Aρ ; hence we have (0, VA λ(A), 0) = V˜A A = AV˜A = (0, 0, (VA∗ λ(A ))∗ ) for all A ∈ Aτ τ . Therefore VA λ(A) = 0 for all A ∈ Aτ τ , which implies that (I − PA )(NAτ τ − NA ) = NAτ τ VA = 0.

(4.1.7)

Thus we get VA NAτ τ = 0. By (4.1.6) and (4.1.7) we have NAτ τ = NA .

(4.1.8)

Thus statement (1) holds by (4.1.5) and (4.1.8). By Lemma 4.1.9 we have PA = I − NAτ , which implies by (4.1.8) that PAτ = I − NAτ τ = I − NA . Hence, statement (2) holds. This completes the proof. Theorem 4.1.7 shows the following Corollary 4.1.12 Let A be a CT ∗ -algebra on H. Then FAτ = FA ,

SAτ = VA , ZAτ = ZA , VAτ = SA .

4.2 Classification of CT ∗ -Algebras In this section we define the notions of regular, semifinite, nondegenerate, singular and nilpotent CT ∗ -algebras and characterize them. Definition 4.2.1 Let A be a CT ∗ -algebra on H. If A = F (A) := τFA (A), then A is called semisimple, and if A = P (A), then A is called nondegenerate. If FA = I , then A is called regular, and if NA = I , then A is called singular. If PA = 0, then A is called nilpotent. We first investigate the relationship among regularity, semisimplicity and nondegenetateness of CT ∗ -algebras.

4 Structure of CT ∗ -Algebras

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Proposition 4.2.2 Let A be a CT ∗ -algebra on H. Consider the following statements: A is regular. A is semisimple. PA = I . A is nondegenerate. Z(A) = {O}.

(i) (ii) (iii) (iv) (v)

(i)

⇒ (ii)

⇐

⇐

Then

(4.2.1)

(iii) ⇒ (iv) ⇔ (v). Each of the converse implications in (4.2.1) don’t necessarily hold. Suppose that λ(A) is dense in H, then (i) and (ii) are equivalent, and (iii) and (iv) are equivalent. Proof The implications (i)⇒(ii) and (i)⇒(iii) are trivial, and (ii)⇒(iv) and (iii)⇒(iv) follow from Theorem 4.1.4. The equivalence of (iv) and (v) follows from Theorem 4.1.4 and Theorem 4.1.6. We give examples in next Example 4.2.3 that each of the converse implications in (4.2.1) don’t hold. It is clear that if λ(A) is dense in H, then (i) and (ii) are equivalent and (iii) and (iv) are equivalent. This completes the proof. Example 4.2.3 (1) Let A0 be a C ∗ -algebra on H which is not nondegenerate. We define a CT ∗ algebra A on H by A = {(A0 , A0 x, (A∗0 x)∗ );

A0 ∈ A0 }

for some x = 0 ∈ H. Since N(A) = {O}, we have FA = PA and τFA (A) = A; hence A is semisimple. However, since π(A) = A0 is not nondegenerate, FA = PA = I , so that A is not regular. Thus, (ii) does not lead to (i). Let {eα0 } be an approximate identity for the C ∗ -algebra A0 and write eα = (eα0 , eα0 x, (eα0 x)∗ ). Then, for any A := (A0 , A0 x, (A∗0 x)∗ ) ∈ A it follows that eα A ∈ P (A) and limeα A = A uniformly, which implies that A = P (A), that is, A is α nondegenerate. Thus, (iv) does not lead to (iii). (2) Let A = T ∗ (H). Since π(A) = B(H) and λ(N(A)) = H, it follows that PA = I and NA = I , so that FA = 0, which implies that (iii)⇒(i) and (iv)⇒(ii) don’t hold.

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79

Definition 4.2.4 For a T ∗ -algebra A on H, the family of subsets {gα } of H satisfying the following conditions will be denoted by MA : (π(A)gα |gβ ) = 0 if α = β and



π(A)gα 2  λ(A)2

α

for all A ∈ A. The main result for the structure of CT ∗ -algebras is Theorem 4.2.6, however it is convenient first to prove the following preliminary important result: Proposition 4.2.5 Let A be a CT ∗ -algebra on H. Then A is regular if and only if Aτ is regular. If A is regular, then there exists an element {gα } of MA such that λ(A) =



π(A)gα

(4.2.2)

α

for all A ∈ A. Proof It follows from Theorem 4.1.7 that FA = PA (I − NA ) = PA PAτ = FAτ , so that A is regular if and only if Aτ is regular. Suppose that A is regular. Then if λ(A) = {0}, then (4.2.2) is trivial. Hence we may assume that A is a regular CT ∗ -algebra with λ(A) = {0}. Then we shall show (4.2.2) in the following process: Step 1 We denote by P the set of all E ∈ Aτ satisfying the following three conditions: π(E) is a projection, λ(E) = 0 and π(τI −π(E) (A)τ ) is nondegenerate. Then P is not empty. Proof For any A ∈ A, T ∈ Aτ and x ∈ H we have (λ(A)|π(T )∗ x) = (π(T )λ(A)|x) = (π(A)λ(T )|x) = (λ(T )|π(A)∗ x). Since λ(A) = {0} by assumption, λ(A) = 0 for some A ∈ A. Since PAτ = I by regularity, there exist T ∈ Aτ and x ∈ H with (λ(A)|π(T )∗ x) = 0. Hence the above equality implies that λ(T ) = 0. Furthermore, since PA = I , there exists a net {eα } in A such that {π(eα )} ⊂ u(π(A)+ ) and converges strongly to I as in (4.1.1). Then we have lim(π(eα )λ(T  T )|λ(eα )) = lim(π(T  T )λ(eα )|λ(eα )) α

α

= lim π(eα )λ(T )2 α

= λ(T )2 = 0,

4 Structure of CT ∗ -Algebras

80

so that λ(T  T ) = π(T )∗ λ(T ) = 0 and π(T  T ) = 0. Thus there exists an element K of Aτ such that π(K)  0, π(K) = 0 and λ(K) = 0.

(4.2.3)

Using the spectral resolution of π(K):

π(K)

π(K) =

t dE(t), 0

we can choose t0 ∈ (0, π(K)) satisfying H0 := π(K)

−1

π(K)

(I − E(t0 )) =

t −1 dE(t) = 0.

(4.2.4)

t0

Since π(K) ∈ π(A) , every element π(A) of π(A) commutes with the spectral projections E(t) of π(K); hence with I − E(t0 ) as well as H0 . We here put E = (I − E(t0 ), H0 λ(K), (H0 λ(K  ))∗ ). Then, for any A ∈ A AE = (π(A)(I − E(t0 )), π(A)H0 λ(K), ((I − E(t0 ))λ(A ))∗ ) = ((I − E(t0 ))π(A), H0 π(K)λ(A), (H0 π(K  )λ(A ))∗ ) = ((I − E(t0 ))π(A), (I − E(t0 ))λ(A), (π(A)∗ H0 λ(K  ))∗ ) = EA; hence, E ∈ Aτ . Furthermore, by (4.2.3) and (4.2.4) E ∈ Aτ , π(E) = I − E(t0 ) and λ(E) = H0 λ(K) = 0.

(4.2.5)

Furthermore, we can prove that π(τI −π(E) (A)τ ) = π(τE(t0 ) (A)τ ) is nondegenerate. Indeed, since (I − E(t0 ), 0, 0)(π(A), E(t0 )λ(A), (E(t0 )λ(A ))∗ ) = ((I − E(t0 ))π(A), 0, 0) = (π(A)(I − E(t0 )), 0, 0) = (π(A), E(t0 )λ(A), (E(t0 )λ(A ))∗ )(I − E(t0 ), 0, 0)

(4.2.6)

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81

and (π(K)E(t0 ), E(t0 )λ(K), (E(t0 )λ(K  ))∗ )(π(A), E(t0 )λ(A), (E(t0 )λ(A ))∗ ) = (π(K)E(t0 )π(A), π(K)E(t0 )λ(A), (π(A)∗ E(t0 )λ(K  ))∗ ) = (π(A)π(K)E(t0), π(A)E(t0 )λ(K), (π(K)∗ E(t0 )λ(A ))∗ ) = (π(A), E(t0 )λ(A), (E(t0 )λ(A ))∗ )(π(K)E(t0 ), E(t0 )λ(K), (E(t0 )λ(K  ))∗ ) for all A ∈ A, it follows that both (I − E(t0 ), 0, 0) and (π(K)E(t0 ), E(t0 )λ(K), (E(t0 )λ(K  ))∗ ) belong to τE(t0 ) (A)τ . Suppose now y ∈ (π(τE(t0 ) (A)τ )H)⊥ . Then, E(t0 )y = y and (x|π(K)y) = (π(K)E(t0 )x|y) = 0 for all x ∈ H. Hence π(K)y = 0 and y = 0 by the existence of π(K)−1 , which implies that π(τE(t0 ) (A)τ ) is nondegenerate. Thus it follows from (4.2.5) and (4.2.6) that E ∈ P. Step 2 Let {Eα }α∈ be a maximal family in P such that {π(Eα )}α∈ is mutually orthogonal and write gα = λ(Eα ) for α ∈ . Then {gα } ∈ MA . Proof Since PAτ = I , there exists a net {eγ } in uπ (Aτ ) such that π(eγ ) converges strongly to I as in (4.1.1). Then we can show that (π(A)gα |gβ ) = (π(A)λ(Eα )|λ(Eβ )) = (λ(A)|λ(Eα Eβ )) = lim(π(eγ )λ(A)|λ(Eα Eβ )) γ

= lim(λ(eγ )|π(Eα )π(Eβ )λ(A )) γ

=0 for all A ∈ A and α = β, and furthermore that  α∈

π(A)gα 2 =



π(Eα )λ(A)2  λ(A)2

α

for all A ∈ A. Thus, {gα } ∈ MA . 0 = I − τ Let E 0 (A) ) is nondegenerate. α∈ π(Eα ). Assume now that π(τE  0 λ(A) = 0, equivalently, λ(A) = We will show that E α∈ π(A)gα for all A ∈ A. After this we assume that there exists an element B of A such that 0 E λ(B) = 0.

(4.2.7)

Step 3 There exists an element S of uπ (τE 0 (A)τ ) such that π(S) = 0 and 0 λ(S) = 0. π(B)E



4 Structure of CT ∗ -Algebras

82

Proof Since π(τE 0 (A)τ ) is nondegenerate, there exists a net {Sα } in uπ (τE 0 (A)τ )   such that π(Sα ) converges strongly to I . Then 0 0 lim π(B)E λ(Sα ) = lim E π(B)λ(Sα ) α

α

0 0 π(Sα )E λ(B) = lim E α

0 = E λ(B)

= 0 by assumption (4.2.7). Thus there exists an element S ∈ uπ (τE 0 (A)τ ) with π(S) = 

0 λ(S) = 0. 0 and π(B)E

0 λ(S). Then there exists an element C of π(A) such that Step 4 Let g = E 0 0 C0 E λ(A) = π(A)g for all A ∈ A, 0 λ(A))⊥ . C0 x = 0 for all x ∈ (E

(4.2.8)

Proof Since S ∈ uπ (τE 0 (A)τ ), we have 

0 0 0 0 λ(S) = E π(S)E λ(A)  E λ(A) π(A)g = π(A)E

for all A ∈ A. Hence our assertion follows. Step 5 Let C0 = U |C0 | be the polar decomposition of C0 and let C = 0 (|C0 |, U ∗ g, (U ∗ g)∗ ) for g = E λ(S). Then C ∈ Aτ . Proof By Step 4, we have 0 0 |C0 |E |C0 | = |C0 |E = |C0 |, 0 , U ∗ U  E

which implies that AC = (π(A)|C0 |, π(A)U ∗ g, (|C0 |λ(A ))∗ ) 0 = (|C0 |π(A), U ∗ π(A)g, (C0 E λ(A ))∗ ) 0 = (|C0 |π(A), U ∗ C0 E λ(A), (π(A)∗ U ∗ g)∗ )

= CA for all A ∈ A. Hence, C ∈ Aτ .

(4.2.9)

4.2 Classification of CT ∗ -Algebras

83

 C  Step 6 Let |C0 | = 0 0 λdF (t) be the spectral resolution of |C0 |. Then there 0 , F := (I − exists t0 ∈ (0, C0 ) such that I − F (t0 ) = 0, I − F (t0 )  E t0 −1 −1 ∗ τ F (t0 ), |C0 | (I − F (t0 ))λ(C), (|C0 | (I − F (t0 ))λ(C)) ) ∈ A and λ(Ft0 ) = 0. 0 λ(B) = π(B)g = 0, it follows from (4.2.8) that |C | = 0 Proof Since U |C0 |E 0 and U ∗ g = 0. Hence there exists t0 ∈ (0, C0 ) such that

I − F (t0 ) = 0.

(4.2.10)

Since C ∈ Aτ , it follows that AFt0 = (π(A)(I − F (t0 )), π(A)|C0 |−1 (I − F (t0 ))λ(C), ((I − F (t0 ))λ(A ))∗ ) = ((I − F (t0 ))π(A), |C0 |−1 (I − F (t0 ))|C0 |λ(A), (|C0 |−1 (I − F (t0 ))π(C)λ(A ))∗ ) = ((I − F (t0 ))π(A), (I − F (t0 ))λ(A), (π(A)∗ |C0 |−1 (I − F (t0 ))λ(C))∗ ) = F t0 A

for all A ∈ A, which means that Ft0 ∈ Aτ .

(4.2.11)

0 I − F (t0 )  E

(4.2.12)

λ(Ft0 ) = |C0 |−1 (I − F (t0 ))λ(C) = 0.

(4.2.13)

Furthermore, we can prove that

and

Indeed, since 0 0 (I − F (t0 )) = E |C0 |(|C0 |−2 |C0 |)(I − F (t0 )) E

= I − F (t0 ) by (4.2.9), we have (4.2.12). Suppose now that λ(Ft0 ) = 0. Then, it follows from (4.2.12) that 0 (I − F (t0 ))E λ(A) = π(Ft0 )λ(A) = π(A)λ(Ft0 ) = 0

for all A ∈ A and (I − F (t0 ))x = |C0 |−1 (I − F (t0 ))|C0 |x = 0

4 Structure of CT ∗ -Algebras

84

0 for all x ∈ (E λ(A))⊥ , which implies that (I − F (t0 )) = 0. This contradicts (4.2.10). Thus, λ(Ft0 ) = 0.   0 Step 7 Let F0 = I − α∈ π(Eα ) + π(Ft0 ) = E − π(Ft0 ). Then

π(τF 0 (A)τ ) is nondegenerate.

(4.2.14)



Proof Take an arbitrary K ∈ τE 0 (A)τ . Then it follows that 

0 π(K)F0 λ(A) = π(K)E λ(A) − π(K)π(Ft0 )λ(A)

= π(A)(λ(K) − π(K)λ(Ft0 )), and that π(K)∗ F0 λ(A) = π(A)(λ(K  ) − π(K)∗ λ(Ft0 )) for all A ∈ A, which implies that 

 π(K), λ(K) − π(K)λ(Ft0 ), (λ(K  ) − π(K)∗ λ(Ft0 ))∗ ∈ τF 0 (A)τ . 

Hence, π(τE 0 (A)τ ) ⊂ π(τF 0 (A)τ ), which yields (4.2.14) since π(τE 0 (A)τ ) is    nondegenerate.  Step 8 λ(A) = α∈ π(A)gα for all A ∈ A. Proof By Step 6, 7, {Eα }∪{Ft0 } is an element of P such that {π(Eα )}α∈ ∪{π(Ft0 )} are mutually orthogonal and τI −α∈ π(Eα )+π(Ft ) (A)τ is nondegenerate. This 0 contradicts to the maximality of {Eα }α∈ . Hence, assumption (4.2.7) is not true, that is,   π(Eα )λ(A) = π(A)gα E λ(A) = 0, equivalently λ(A) = α

α

for all A ∈ A. This completes the proof of Proposition 4.2.5. The following theorem for the semisimplicity of CT ∗ -algebras is one of the most important results of this note: Theorem 4.2.6 Let A be a CT ∗ -algebra on H. Then the following statements are equivalent: (i) (ii) (iii) (iv)

A is semisimple. πA is an injection. N(A) = {O}. π(Aτ ) is nondegenerate.

4.2 Classification of CT ∗ -Algebras

85

(v) There exists an element {gα } of MA such that λ(A) =



π(A)gα

α

for all A ∈ A. λ(A) = sup{π(A)λ(K); K ∈ uπ (Aτ )} for all A ∈ A. The map: π(A) → λ(A) of π(A)[τuπ ] into the Hilbert space H is closable. The map: π(A) → λ(A) of π(A)[τσπs ∗ ] into the Hilbert space H is closable. The map: π(A) → ν(A) of π(A)[τuπ ] into the Hilbert space N(T ∗ (H)) is closable. (vv) The map: π(A) → ν(A) of π(A)[τσπs ∗ ] into the Hilbert space N(T ∗ (H)) is closable.

(vi) (vii) (viii) (viv)

In (vii)–(vv) τuπ and τσπs ∗ are the uniform topology and the σ s ∗ -topology on π(A), respectively, and the Hilbert space H and N(T ∗ (H)) are equipped with the norm topologies. Proof The implications (i)⇒(iii)⇒(ii) are trivial. (ii)⇒(i) For any A ∈ A we have A − τFA (A) = (0, λ(A) − FA λ(A), (λ(A ) − FA λ(A ))∗ ), which implies by (ii) that λ(A) = FA λ(A) and λ(A ) = FA λ(A ). Hence A = τFA (A), which means that A is semisimple. The equivalence of (iii) and (iv) follows from Theorem 4.1.7. Thus, (i)–(iv) are equivalent. (i)⇒(v) By the semisimplicity of A, we have FA = PA , π(A)PA = π(A) and PA λ(A) = λ(A) for all A ∈ A, which implies that APA := {(π(A)PA H , λ(A), λ(A )∗ ); A ∈ A} is a regular CT ∗ -algebra on PA H. Hence it follows from Proposition 4.2.5 that there exists an element {gα } of MAPA such that λ(A) =



π(A)gα

α

for all A ∈ A.Furthermore, it is easily shown that MPA ⊂ MA , so that (v) holds.

4 Structure of CT ∗ -Algebras

86

(v)⇒(ii) This is trivial. Thus, (i)–(v) are equivalent. (iv)⇒(vi) By assumption (iv) there exists a net {eγ } in uπ (Aτ ) such that π(eγ ) converges strongly to I as in (4.1.1). Then we have lim π(A)λ(eγ ) = lim π(eγ )λ(A) γ

γ

= λ(A) for all A ∈ A, which implies (vi). (vi)⇒(ii) This is trivial. (iv)⇒(viii) Suppose that {Aα } is any net in A such that {π(Aα )} converges σ strongly∗ to 0 and {λ(Aα )} converges to some element x of H. Since π(Aτ ) is nondegenerate, it follows that x = lim π(eγ )x γ

= lim lim π(eγ )λ(Aα ) γ

α

= lim lim π(Aα )λ(eγ ) γ

α

= 0. Thus, π(A) → λ(A) is σ -strongly∗ closable. (viii)⇒(vii) This is trivial. (viii)⇒(vv)⇒(viv) This is trivial. (vii)⇒(iii) Take an arbitrary A ∈ N(A), and put An = A, n ∈ N. Then since π(An ) = 0, λ(An ) = λ(A) and λ(An ) = λ(A ) for any n ∈ N, it follows from (vi) that λ(A) = λ(A ) = 0. Hence, A = O. Similarly we can prove (viv)⇒(iii). Thus, (i)–(vv) are equivalent, which completes the proof. Since τFA (A) is a semisimple CT ∗ -algebra on H for every CT ∗ -algebra A on H, Theorem 4.2.6 proves the following Corollary 4.2.7 Let A be a CT ∗ -algebra on H. Then there exists an element {gα } of MAFA such that FA λ(A) =

 α

π(A)gα

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87

for all A ∈ A, and FA λ(A)2 =



π(A)gα 2

α

= sup{π(A)λ(K);

K ∈ uπ (τFA (A)τ )}

for all A ∈ A. For the commutants and bicommutants of semisimple CT ∗ -algebra we have the following Theorem 4.2.8 Suppose that A is a semisimple CT ∗ -algebra on H. Then Aτ = Aρ ,

Aρρ ⊂ Aτ τ ⊂ Aππ and AT = Aρρ .

In particular, if A is regular, then AT = Aτ τ = Aρρ . Proof For any S, K ∈ Aτ and C0 ∈ π(A) we have   A(S C˜0 K) = π(A)π(S)C0 π(K), π(A)π(S)C0 λ(K), (π(K)∗ C0∗ π(S)∗ λ(A ))∗   = π(S)C0 π(K)π(A), π(S)C0 π(K)λ(A), (π(A)∗ π(K)∗ C0∗ λ(S  ))∗ = (S C˜0 K)A for all A ∈ A, where C˜0 := (C0 , 0, 0). Hence S C˜ 0 K ∈ Aτ . Furthermore, we get that (λ(S C˜0 K)|λ(A )) = (π(S)C0 λ(K)|λ(A )) = (C0 λ(K)|π(A)∗ λ(S  )) = (C0 π(K)λ(A)|λ(S  )) = (λ(A)|π(K)∗ C0∗ λ(S  )) = (λ(A)|λ((S C˜0 K) )) for all A ∈ A. Hence it follows that S C˜0 K ∈ Aρ

(4.2.15)

for all S, K ∈ Aτ and C0 ∈ π(A) . We show that Aτ τ ⊂ Aππ .

(4.2.16)

4 Structure of CT ∗ -Algebras

88

Indeed, since Aππ = {A ∈ T ∗ (H); π(A) ∈ π(A) }, it suffices to show that τ τ π(A) ∈ π(A) for all A ∈ A . Take an arbitrary A ∈ Aτ τ . Since π(Aτ ) is nondegenerate by Theorem 4.2.6, as seen often, there exists a net {eγ } in uπ (Aτ ) such that π(eγ ) converges strongly to I . By (4.2.15) we have (eγ ) C˜0 eγ ∈ Aρ ⊂ Aτ for all γ and C0 ∈ π(A) , so that π(eγ )∗ C0 π(eγ )π(A) = π(A)π(eγ )∗ C0 π(eγ ),     lim π(eγ )∗ C0 π(eγ )π(A)x|y = lim C0 π(eγ )π(A)x|π(eγ )y γ

γ

= (C0 π(A)x|y) and   lim π(A)π(eγ )∗ C0 π(eγ )x|y = (C0 x|π(A)∗ y) γ

for all A ∈ A and x, y ∈ H. Hence, π(A) ∈ π(A) . Thus, (4.2.16) holds. Since PAτ = I , it follows from Proposition 4.2.2 that Aτ = P (Aτ ), which implies by (4.2.15) that Aτ = P (Aτ ) ⊂ Aρ . Therefore Aτ = Aρ and Aρρ = Aτρ ⊂ Aτ τ .

(4.2.17)

By (4.2.16) and (4.2.17) we get Aρρ ⊂ Aτ τ ⊂ Aππ and AT = Aρρ . Suppose now that A is regular. By Proposition 4.2.5, Aτ is regular, so that it follows from (4.2.17) that Aτ τ = Aτρ = Aρρ . This completes the proof. Remark By Proposition 4.2.5, A is regular if and only if Aτ is regular, but we don’t know the relationship between the semisimplicity of A and of Aτ . For the singularlity of A we have the following Theorem 4.2.9 Let A be a CT ∗ -algebra on H. Then the following statements hold: (1) A is singular if and only if for any x ∈ H there exists a sequence {An } in N(A) such that lim λ(An ) = x. n→∞

4.2 Classification of CT ∗ -Algebras

89

(2) A is singular and nondegenerate if and only if SA = I . In this case, we have Aτ = {O} and AT = Aππ . (3) Suppose that A is nondegenerate. Then the following (i)–(iv) are equivalent: (i) A is singular. (ii) For any x, y ∈ H there exists an element A of N(A) such that λ(A) = x and λ(A ) = y. (iii) π(A) is a C ∗ -algebra on H, and (π(A), 0, 0) ⊂ A, N(T ∗ (H)) ⊂ A and A = (π(A), 0, 0) + N(T ∗ (H)). (iv) λ(N(A)) = H. Proof (1) This is trivial. (2) Suppose that A is nondegenerate and singular. Then, since NA = I , it follows that λ(N(A)) is dense in H, which implies by the nondegenerateness of A that PA = I . Thus, SA = I . The converse is trivial. Suppose, conversely, that SA = I . Take an arbitrary K ∈ Aτ . Since PAτ = I −NA = 0 by Theorem 4.1.7, we have π(K) = 0. Furthermore, since π(A)λ(K) = π(K)λ(A) = 0 and π(A)λ(K  ) = 0 for all A ∈ A, it follows from PA = I that λ(K) = λ(K  ) = 0. Hence, Aτ = {O}. Since Aρ ⊂ Aτ , we have Aρ = {O}, which implies that Aτ τ = Aρρ = T ∗ (H). Therefore AT = Aππ . (3) The implications (iii)⇒(iv)⇒(i) are trivial. (i)⇒(ii) Since SA = I by (2), it follows from Lemma 4.1.5, (1) that (0, x, y ∗ ) ∈ N(SA ) = P (A) ∩ N(A) ⊂ A for all x, y ∈ H, which proves (ii). (ii)⇒(iii) Take an arbitrary A ∈ A. By assumption (ii) we have (0, λ(A), λ(A )∗ ) ∈ A and (π(A), 0, 0) = A − (0, λ(A), λ(A )∗ ) ∈ A.

(4.2.18)

4 Structure of CT ∗ -Algebras

90

Thus, (π(A), 0, 0) ⊂ A. Take an arbitrary sequence {An } in A such that lim π(An ) − X0  = 0 for some X0 ∈ B(H). Then, since n→∞

lim (π(An ), 0, 0) − (X0 , 0, 0) = 0

n→∞

and A is a CT ∗ -algebra, it follows from (4.2.18) that X := (X0 , 0, 0) ∈ A and X0 = π(X) ∈ π(A). Thus π(A) is a C ∗ -algebra on H, which proves (iii). This completes the proof. We next characterize nilpotent CT ∗ -algebras. Proposition 4.2.10 Let A be a CT ∗ -algebra on H. Then the following statements are equivalent: (i) (ii) (iii) (iv)

A is nilpotent. A = N(A). A = ν(A). A = Z(A).

Proof The equivalence of (i), (ii) and (iii) is almost trivial. (i)⇒(iv) By Theorem 4.1.4 and (iii) we have Z(A) = νI −PA (A) = ν(A) = A. (iv)⇒(ii) Since Z(A) = νI −PA (A) ⊂ N(A) ⊂ A = Z(A), it follows that N(A) = A and PA = 0. Hence, A is nilpotent. This completes the proof. We finally characterize nilpotent and singular CT ∗ -algebras. For that, we regard ν(A) = {(0, λ(A), λ(A )∗ );

A ∈ A}

as a subspace of the Hilbert space H ⊕ H ∗ with inner product: 

 (x1 , y1∗ )|(x2 , y2∗ ) = (x1 |x2 ) + (y2 |y1 ),

and denote by ν(A)⊥ the orthogonal complement of ν(A) in H ⊕ H ∗ . Lemma 4.2.11 Suppose that A is a nilpotent CT ∗ -algebra on H. Then ν(A) is a closed subspace in the Hilbert space H ⊕ H ∗ and ν(Aρ ) = J ν(A)⊥ , where  J =

I 0 0 −I

 .

4.2 Classification of CT ∗ -Algebras

91

Proof By Lemma 2.6.7 we have ν(Aρ ) ⊂ J ν(A)⊥ . We show the converse. Take an arbitrary (x, y ∗ ) ∈ ν(A)⊥ , and write K = (0, x, −y ∗ ). Then it is easily shown that K ∈ Aτ and (λ(A)|x) = (−y|λ(A )) for all A ∈ A. Hence, K ∈ Aρ and J (x, y ∗ ) = ν(K) ∈ ν(Aρ ). Thus, J ν(A)⊥ ⊂ ν(Aρ ). This completes the proof. Proposition 4.2.12 Let A be a CT ∗ -algebra on H. Then the following statements are equivalent: (i) (i) (ii) (ii)

A is singular and nilpotent. Aτ is singular and nilpotent. A is nilpotent and λ(A) is dense in H. Aτ is nilpotent and λ(Aτ ) is dense in H.

In this case, then AT = A. Proof The equivalence of (i) and (ii) is trivial. Furthermore, the equality: ZA = (I − PA )NA = NAτ (I − PAτ ) = ZAτ implies that statements (i), (ii), (i) and (ii) are equivalent. Suppose that A is singular and nilpotent. Then A = N(A) and λ(A) is dense in H by Proposition 4.2.9, so that Aπ = T ∗ (H) and Aππ = N(T ∗ (H))

(4.2.19)

Aτ = N(T ∗ (H)).

(4.2.20)

Aρ = ν(Aρ ) = N(Aρ ).

(4.2.21)

and

Therefore

Furthermore, it follows from Lemma 4.2.10 that Aρ = ν(Aρ ) = J ν(A)⊥ and ν(Aρρ ) = J ν(Aρ )⊥ = J (J ν(A)⊥ )⊥ = ν(A)⊥⊥ = ν(A) = A.

(4.2.22)

4 Structure of CT ∗ -Algebras

92

Since Aτ is singular and nilpotent, it follows, as in (4.2.20), that Aτ τ = N(T ∗ (H)).

(4.2.23)

By (4.2.19), (4.2.22) and (4.2.23) we have AT = N(T ∗ (H)) ∩ Aρρ = ν(Aρρ ) = A. This completes the proof. In case that a CT ∗ -algebra A on H is isomorphic to a CT ∗ -algebra B on K, it is natural to consider whether the regularity (resp. the semisimplicity, the nondegenerateness, the singularity, the nilpotentness) of A and B is equivalent. For this question we get the following Proposition 4.2.13 Let A be a CT ∗ -algebra on H and B a CT ∗ -algebra on K. Suppose that A is isomorphic to B. Then A is semisimple (resp. nondegenerate, singular, nilpotent) if and only if B is semisimple (resp. nondegenerate, singular, nilpotent). However, the regularity of A and B is not necessarily equivalent. Proof Let  be an isometric ∗-isomorphism and isometry of A[τu ] onto B[τu ]. Since π((A)) = π(A) and

λ((A)) = λ(A)

for all A ∈ A, it follows that P (B) = (P (A)) and N(B) = (N(A)), which implies that the semisimplicity (resp. nondegenerateness, singularity, nilpotentness) of A and B is equivalent. In the next Example 4.2.14 we give a semisimple and nonregular CT ∗ -algebra which is isomorphic to a regular CT ∗ -algebra. This completes the proof. Example 4.2.14 Let A be a CT ∗ -algebra on H such that 0 < = PA < = I . Then ∗ τPA (A) is a CT -algebra on H which is not regular. We define a T ∗ -algebra APA on PA H by  &  APA = APA := π(A)PA H , PA λ(A), (PA λ(A ))∗ ;

' A∈A .

Then it is easily shown that the map  defined by :

A ∈ τPA (A) −→ APA ∈ APA

is a ∗-isomorphism of the CT ∗ -algebra τPA (A) onto the T ∗ -algebra APA . Furthermore, since π(A) = sup{π(A)x; x  1} = sup{π(A)PA x;

x  1}

4.3 Commutative Semisimple CT ∗ -Algebras

 sup{π(A)PA x;

93

PA x  1}

= π(A)PA H   π(A) for all A ∈ A, it follows that  is an isometry. Hence, APA is a regular CT ∗ -algebra on PA H which is isomorphic to the nonregular CT ∗ -algebra τPA (A) on H.

4.3 Commutative Semisimple CT ∗ -Algebras In this section we shall show that every commutative semisimple CT ∗ -algebra A[τu ]   ∗ 2 is isomorphic to a regular CT -algebra C0 () ∩ L (, μ) [τu ]. Here C0 () is the commutative C ∗ -algebra of continuous functions f on a locally compact space  vanishing at infinity on  equipped with the usual function operations: f + g, αf , fg, the involution f ∗ (f ∗ (λ) = f (λ)) and the C ∗ -norm f u := supx∈ |f (x)|, and L2 (, μ) is the L2 -space defined by a regular Borel measure μ on . Then C0 () ∩ L2 (, μ) is a Banach ∗-algebra with the above function operations f + g, αf , fg, the involution f ∗ and the norm f  := max(f u , f 2 ), and it is regarded as the regular commutative CT ∗ -algebra:   τ C0 () ∩ L2 (, μ) := {τ (f ) = (π(f ), λ(f ), λ(f ∗ )∗ ); f ∈ C0 () ∩ L2 (, μ)},

where π(f ) is a bounded operator on L2 (, μ) defined by π(f )g = fg, g ∈ L2 (, μ) and λ(f ) = f . Then we get the following main theorem of this section. Theorem 4.3.1 Let A be a commutative semisimple CT ∗ -algebra on H. Then there exist a locally compact space  and a regular Borel measure μ on  such that   A[τ ] is isomorphic to the commutative regular CT ∗ -algebra C0 () ∩ L2 (, μ) [τu ]. Since π(A)[τuπ ] of the uniform closure of π(A) is a commutative C ∗ -algebra, it follows from ([37, Theorem 4.4]) that there exists a locally compact space  such that it is isomorphic to the commutative C ∗ -algebra C0 (). Denote the isometric ∗isomorphism of π(A)[τuπ ] onto C0 () by ϕ. Let Cc () be the subset of continuous functions with compact support on . The support of a function f is denoted by supp f . We will prove Theorem 4.3.1 in the following process. Step 1 Cc () ⊂ ϕ(π(A)). Proof For any f ∈ C0 () there exists a sequence {An } in A such that lim π(An ) − ϕ −1 (f ) = 0. Since A is commutative, we have

n→∞

π(B)∗ λ(An ) = λ(B  An ) = λ(An B  ) = π(A∗n )λ(B  )

4 Structure of CT ∗ -Algebras

94

for all B ∈ A. Hence it follows that   An B = π(An )π(B), π(An )λ(B), (π(B)∗ λ(An ))∗   = π(An )π(B), π(An )λ(B), (π(An )∗ λ(B  ))∗ , and   τu - lim An B = ϕ −1 (f )π(B), ϕ −1 (f )λ(B), (ϕ −1 (f )∗ λ(B  ))∗ . n→∞

The right hand side can be interpreted as an action of ϕ −1 (C0 ()) to A, so that it will be denoted by ϕ −1 (f )B. Therefore, ϕ −1 (f )A ⊂ P (A)

(4.3.1)

for all f ∈ C0 (). For any x ∈  there exists an element A of A such that ϕ(π(A))(x) = 0, so ϕ(π(A A)(x) = |ϕ(π(A))x|2 > 0. Hence, for any f ∈ Cc () there exists an element B of A such that ϕ(π(B))(x) > 0

on

x ∈ supp f,

ϕ(π(B))(x)  0

on

x ∈  \ supp f,

so that there exists an element h of Cc () such that h(x) = (ϕ(π(B))(x))−1 for all x ∈ supp f . Let A = ϕ −1 (g)B. Then A ∈ A by (4.3.1) and f = f hϕ(π(B)) = ϕ(ϕ −1 (f h)π(B)) = ϕ(π(A)) ∈ ϕ(π(A)). In what follows we denote the set {A ∈ A; ϕ(π(A)) ∈ Cc ()} by Ac . Then the mapping :

A ∈ Ac −→ ϕ(π(A)) ∈ Cc ()

is surjective by Step 1, and is injective by the semisimplicity of A. Furthermore, we can easily show the following Step 2 Ac is a ∗-subalgebra of the CT ∗ -algebra A, Cc () is a ∗-subalgebra of the CT ∗ -algebra C0 () ∩ L2 (, μ) and  is an algebraic ∗-isomoprphism of the T ∗ -algebra Ac onto the T ∗ -algebra Cc ().

4.3 Commutative Semisimple CT ∗ -Algebras

95

We will verify that  is an isometry of the T ∗ -algebra Ac [τu ] on the T ∗ -algebra Cc ()[τu ]. Since (A) = max (ϕ(π(A))u , ϕ(π(A))2 ) ,   A = max π(A), λ(A), λ(A ) and ϕ(π(A))u = π(A) for all A ∈ Ac , it suffices to show that ϕ(π(A))2 = λ(A). We will prove this equality after some preparations. Step 3 For any A ∈ Ac there exists an element B of Ac such that BA = A and B  A = A. Proof For any A ∈ Ac there exists an element B of Ac such that ϕ(π(B))(x) = 1 for all x ∈ supp ϕ(π(A)). Then ϕ(π(BA)) = ϕ(π(B))ϕ(π(A)) = ϕ(π(A)); hence π(BA) = π(A). Therefore, for any C ∈ Aτ and x ∈ H, we see that (π(B)λ(A)|π(C)x) = (π(BA)λ(C  )|x) = (π(A)λ(C  )|x) = (λ(A)|π(C)x), which implies since π(Aτ ) is nondegenerate that π(B)λ(A) = λ(A). Similarly, π(B)∗ λ(A ) = λ(A ). Hence we have BA = (π(BA), π(B)λ(A), (π(B)∗ λ(A ))∗ ) = (π(A), λ(A), λ(A )∗ ) = A. Since ϕ(π(B  ))(x) = ϕ(π(B))(x), it follows that ϕ(π(B))(x) = 1 for x ∈ supp ϕ(π(A)). Hence we can show B  A = A in the same way as above. Step 4 We define a function F on Cc () by F (f ) = (λ(ϕ −1 (f )B)|λ(B)), f ∈ Cc ().

4 Structure of CT ∗ -Algebras

96

Then it is determined independently from the choice of B ∈ Ac which satisfies ϕ(π(B))(x) = 1 for all x ∈ supp f . Moreover, it is a positive linear functional on Cc (). Proof Let f ∈ Cc (). By Step 1 f = ϕ(π(A)) for some A ∈ Ac . Take arbitrary B, B1 ∈ Ac satisfying ϕ(π(B))(x) = ϕ(π(B1 ))(x) = 1 for all x ∈ supp f , and take an element h ∈ Cc () which h(x) = 1 for all x ∈ supp ϕ(π(B))∪ supp ϕ(π(B1 )). Then, h = ϕ(π(C)) for some C ∈ Ac , so that it follows from Step 3 that 

BA = B  A = A, B1 A = B1 A = A, BC = B

and B1 C = B1 .

(4.3.2)

Hence we have (λ(ϕ −1 (f )B)|λ(B)) = (λ(AB)|λ(B)) = (λ(A)|λ(BC)) = (λ(B  A)|λ(C)) = (λ(A)|λ(C)). Similarly, (λ(ϕ −1 (f )B1 )|λ(B1 )) = (λ(A)|λ(C)). Thus, the value of F (f ) does not depend on the choice of B. We next show that F is a positive linear functional on Cc (). Take arbitrary f, g ∈ Cc () and take an element h of Cc () with h = 1 on supp f ∪ supp g∪ supp (f + g). Since f = ϕ(π(A)), g = ϕ(π(B)) and h = ϕ(π(C)) for some A, B, C ∈ Ac , we see that F (f + g) = (ϕ −1 (f + g)λ(C)|λ(C)) = (ϕ −1 (f )λ(C)|λ(C)) + (ϕ −1 (g)λ(C)|λ(C)) = F (f ) + F (g), F (αf ) = (ϕ −1 (αf )λ(C)|λ(C)) = α(ϕ −1 (f )λ(C)|λ(C)) = αF (f ), ∗

F (f f ) = (ϕ −1 (f ∗ f )λ(C)|λ(C)) = (ϕ −1 (f ∗ )ϕ −1 (f )λ(C)|λ(C)) = ϕ −1 (f )λ(C)2  0 fow all f, g ∈ Cc () and α ∈ C. Thus F is a positive linear functional on Cc (). Step 5 ϕ(π(A))2 = λ(A) for all A ∈ Ac .

4.3 Commutative Semisimple CT ∗ -Algebras

97

Proof For any A, B ∈ Ac choose an element C of Ac with ϕ(π(C)) = 1 on supp ϕ(π(A))∪ supp ϕ(π(B))∪ supp ϕ(π(B  A)). Then it follows from (4.3.2) and the Riesz-Markov theorem [29] that (λ(A)|λ(B)) = (λ(AC)|λ(BC)) = (π(B  A)λ(C)|λ(C)) = F (ϕ(π(B  A)))

= ϕ(π(B  A))(λ)dμ(λ) 

=

ϕ(π(B))(λ)ϕ(π(A))(λ)dμ(λ) 

= (ϕ(π(A))|ϕ(π(B))), which completes the proof. Thus,  is an isometric ∗-isomorphism of the T ∗ -algebra Ac [τu ] onto the T ∗ algebra Cc ()[τu ]. Proof of Theorem 4.3.1 We show that  can be extended to the ∗-isomorphism of  the CT ∗ -algebra A[τu ] onto the CT ∗ -algebra C0 () ∩ L2 (, μ) [τu ]. By Step ¯ c [τu ] = A and Cc ()[τu ] = C0 () ∩ L2 (, μ). 2, 4 it suffices to show that A Indeed, take arbitrary A, B ∈ A. Then, π(A) = ϕ −1 (f ) and π(B) = ϕ −1 (g) for some f, g ∈ C0 (). Now, we can take a sequence {fn } in Cc () with lim fn − f u = 0, and then there exists a sequence {An } in Ac with π(An ) = n→∞ ϕ −1 (fn )

and lim π(An ) − π(A) = lim fn − f u = 0.

n→∞

n→∞

Since {fn g} is a sequence in Cc (), it follows that {An B} ⊂ Ac and

lim An B − AB = 0,

n→∞

which implies that Ac is τu -dense in P (A). By the semisimplicity of A, we have ¯ c [τu ] = A. On the other hand it is clear that Cc ()[τu ] = A = P (A); hence A 2 C0 () ∩ L (, μ). Thus the CT ∗ -algebra A[τu ] is isomorphic to the CT ∗ -algebra (C0 () ∩ L2 (, μ))[τu ]. This completes the proof. The next result is now immediate from Theorem 4.3.1. Corollary 4.3.2 Suppose that A is a commutative CT ∗ -algebra on H. Then there exists a locally compact space  and a regular Borel measure μ on  such that the CT ∗ -algebra τFA (A) is isomorphic to the CT ∗ -algebra (C0 () ∩ L2 (, μ)).

4 Structure of CT ∗ -Algebras

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4.4 Some Results Obtained from T ∗ -Algebras 4.4.1 Projections Defined by T ∗ -Algebras ¯ u ] of A by A ˜ for Let A be a T ∗ -algebra on H and denote the uniform closure A[τ ∗ ˜ simplicity. Then A is a CT -algebra on H satisfying ˜ P (A) = P (A),

˜ and Z(A) ⊂ Z(A). ˜ N(A) ⊂ N(A)

˜ we have Hence, for the relationship between the projections defined by A and A PA = PA˜ , NA  NA˜ ,

FA := PA (I − NA )  FA˜ ,

(4.4.1)

SA := PA NA  SA˜ , ZA := (I − PA )NA  ZA˜ , VA := (I − PA )(I − NA )  VA˜ . ˜ τ , it follows from Theorem 4.1.7 and (4.4.1) that Furthermore, since Aτ = A PAτ = PA˜ τ = I − NA˜  I − NA ,

NAτ = NA˜ τ = I − PA˜ = I − PA ,

(4.4.2)

FAτ = FA˜ τ = FA˜  FA , SAτ = SA˜ τ = VA˜  VA , ZAτ = ZA˜ τ = ZA˜  ZA ,

VAτ = VA˜ τ = SA˜  SA .

We now define the notions of semisimplicity, regularity and singularity of T ∗ algebras. Definition 4.4.1 A T ∗ -algebra A is called nondegenerate (resp. semisimple, regular, singular, nilpotent) if PA = I (resp. PAτ = I , PA = PAτ = I , PAτ = 0, PA = 0). From (4.4.1) and (4.4.2) it follows that A is nondegenerate (resp. semisimple, ˜ is nondegenerate (resp. semisimple, regular, regular, singular) if and only if A singular).

4.4.2 The Vector Representation of the CT ∗ -Algebra Generated by a T ∗ -Algebra In this subsection we investigate the relationship between the vector representation ˜ λ(A) of a T ∗ -algebra A and that of the CT ∗ -algebra A.

4.4 Some Results Obtained from T ∗ -Algebras

99

˜ we put Theorem 4.4.2 Let A be a T ∗ -algebra on H. For A ∈ A  F(A) = {An } ⊂ A;

∞ 

π(An )2 < ∞

n=1 ∞ 

and

(

∗

∗

π(An ) π(An )  π(A) π(A) .

n=1

Then, F(A) = ∅ and  2

FA˜ λ(A) = inf

∞ 

( 2

FA˜ λ(An ) ;

{An } ∈ F(A) .

n=1

In particular, if A is semisimple, then  λ(A) = inf 2

∞ 

( λ(An ) ; {An } ∈ F(A) . 2

n=1

We prove this theorem in the following processes. ˜ and {An } ∈ F(A) we have Step 1 For any A ∈ A ∞ 

FA˜ λ(An )2  FA˜ λ(A)2 .

n=1

˜ and {An } ∈ F(A). Then it follows from Corollary 4.2.7 that Proof Let A ∈ A ∞ 

FA˜ λ(An )2 

n=1

∞ 

π(An )λ(K)2

n=1

 π(A)λ(K)2   ˜ τ , which implies by Corollary 4.2.7 again that for all K ∈ uπ τFA˜ (A) ∞  n=1

FA˜ λ(An )2  FA˜ λ(A)2 .

4 Structure of CT ∗ -Algebras

100

˜ and ε > 0, From Step 1 the proof of Theorem 4.4.2 completes if for any A ∈ A we can find an element {An } of F(A) such that FA˜ λ(A)2 + ε >

∞ 

FA˜ λ(An )2 .

n=1

In the following we use the following inequality several times. Let a and b be elements of a ∗-algebra and δ a positive scalar. Since a = b + (a − b), it follows that  1 ∗  1  1 1 a ∗ a  (b + (a − b))∗ (b + (a − b)) + δ − 2 b − δ − 2 (a − b) δ 2 b − δ − 2 (a − b) = (1 + δ)b∗ b + (1 + δ −1 )(a − b)∗ (a − b).

(4.4.3)

In particular, if δ = 1, then a ∗ a  2(b∗b + (a − b)∗ (a − b)).

(4.4.4)

˜ and ε > 0. Take arbitrary A ∈ A ˜ Step 2 There exist a first approximation A1 ∈ A of A and the rest B1 ∈ τFA˜ (A) satisfying (1) π(A)∗ π(A)  π(A1 )∗ π(A1 ) + π(B1 )∗ π(B1 ), (2) π(A1 )2  π(A)2 + ε2 and π(B1 )2 < 2ε5 , (3) FA˜ λ(A1 )2  FA˜ λ(A)2 + ε2 and λ(B1 )2
0 we choose X1 ∈ A with τFA˜ (A − X1 ) < ε1 , namely π(A) − π(X1 ) < ε1 ,

(4.4.5)

FA˜ λ(A) − FA˜ λ(X1 ) < ε1 .

(4.4.6)

We here put As = τFA˜ (A) and (X1 )s = τFA˜ (X1 ), and for any δ > 0, 1

1

A1 = (1 + δ) 2 X1 and B1 = (1 + δ −1 ) 2 (As − (X1 )s ).

4.4 Some Results Obtained from T ∗ -Algebras

101

˜ Since As = (X1 )s +(As −(X1 )s ), π(As ) = π(A) Then, A1 ∈ A, B1 ∈ τFA˜ (A). and π((X1 )s ) = π(X1 ), it follows from (4.4.3) that π(A)∗ π(A)  (1 + δ)π(X1 )∗ π(X1 ) + (1 + δ −1 )π(As − (X1 )s )∗ π(As − (X1 )s ) = π(A1 )∗ π(A1 ) + π(B1 )∗ π(B1 ).

(2) Since X1 = A + (X1 − A), we have π(X1 )∗ π(X1 )  (1 + δ)π(A)∗ π(A) + (1 + δ −1 )π(X1 − A)∗ π(X1 − A). 1

Since A1 = (1 + δ) 2 X1 , it follows from (4.4.5) that π(A1 )∗ π(A1 ) = (1 + δ)π(X1 )∗ π(X1 )  (1 + δ)2 π(A)∗ π(A) + (1 + δ)(1 + δ −1 )π(X1 − A)∗ π(X1 − A)   < π(A)∗ π(A) + (2δ + δ 2 )π(A)∗ π(A) + (1 + δ)(1 + δ −1 )ε12 I ,

and that π(B1 )2 = (1 + δ −1 )π(A) − π(X1 )2  (1 + δ −1 )ε12 . We now choose ε1 and δ as follows:   ε (2δ + δ 2 ) max π(A)2 , FA˜ λ(A)2 + (1 + δ)(1 + δ −1 )ε12 < , 2 (1 + δ −1 )ε12