Generalized B*-Algebras and Applications (Lecture Notes in Mathematics, 2298) [1st ed. 2022] 3030964329, 9783030964320

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Lecture Notes in Mathematics  2298

Maria Fragoulopoulou Atsushi Inoue Martin Weigt Ioannis Zarakas

Generalized B*-Algebras and Applications

Lecture Notes in Mathematics Volume 2298

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 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 László Székelyhidi , Institute of Mathematics, Leipzig University, Leipzig, 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

This series reports on new developments in all areas of mathematics and their applications - quickly, informally and at a high level. Mathematical texts analysing new developments in modelling and numerical simulation are welcome. The type of material considered for publication includes: 1. Research monographs 2. Lectures on a new field or presentations of a new angle in a classical field 3. Summer schools and intensive courses on topics of current research. Texts which are out of print but still in demand may also be considered if they fall within these categories. The timeliness of a manuscript is sometimes more important than its form, which may be preliminary or tentative. Titles from this series are indexed by Scopus, Web of Science, Mathematical Reviews, and zbMATH.

Maria Fragoulopoulou • Atsushi Inoue • Martin Weigt • Ioannis Zarakas

Generalized B*-Algebras and Applications

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

Atsushi Inoue Department of Applied Mathematics Fukuoka University Fukuoka, Japan

Martin Weigt Department of Mathematics and Applied Mathematics Nelson Mandela University Port Elizabeth, South Africa

Ioannis Zarakas Department of Mathematics Hellenic Army Academy Vari, Greece

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-030-96432-0 ISBN 978-3-030-96433-7 (eBook) https://doi.org/10.1007/978-3-030-96433-7 Mathematics Subject Classification: 46H20, 46H35, 46K10, 47L60, 46H30, 46K05, 46L60 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 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

see [G.S. Kirk, J.E. Raven, The Presocratic Philosophers (Cambridge Univ. Press,1957/ with corrections, Cambridge, 1973), pp. 236–237]

Contemporaneously with these philosophers [sc. Leucippus and Democritus], and before them, the Pythagoreans, as they are called, devoted themselves to mathematics; they were the first to advance this study, and having been brought up in it they thought its principles were the principles of all things. W. Jaeger (ed.) Aristotelis Metaphysica book 1, 985b 23-26 see [W. Jaeger (ed.), Aristotelis Metaphysica (Oxford Univ. Press, Oxford, 1957), pp. 13,14]

To the fond memory of the late Professor G.R. ALLAN and to Professor P.G. DIXON with much respect

Preface

In 1967, G.R. Allan initiated and studied a class of locally convex algebras with continuous involution, called GB∗ -algebras (an abbreviation for “Generalized B ∗ algebras”). The structure of a GB∗ -algebra A[τ ] is defined by a certain collection B∗A of subsets of the underlying locally convex ∗-algebra A[τ ], which, being partially ordered by inclusion, attains a maximal member, denoted by B0 . Among the first results, Allan showed an algebraic Gelfand–Naimark type theorem for commutative GB∗ -algebras. Namely, he proved that a commutative GB∗ -algebra A[τ ] is algebraically ∗-isomorphic to a ∗-algebra of extended-complex valued continuous functions on a compact Hausdorff space M0 . The latter is, in fact, the Gelfand space of the C ∗ -subalgebra A[B0 ] := {λx : λ ∈ C, x ∈ B0 } of A[τ ], endowed with the gauge function of B0 . This C ∗ -algebra is the key tool for investigating the structure of a GB∗ -algebra. In 1970, P.G. Dixon extended Allan’s definition of a GB∗ -algebra to include topological ∗-algebras that are not locally convex, thus enriching the set of examples of GB∗ -algebras. Dixon then showed that the locally convex noncommutative GB∗ algebras are realized by closed operators on a Hilbert space. Namely, he gave a noncommutative algebraic Gelfand–Naimark type theorem for GB∗ -algebras, which determines them among unbounded operator algebras. Typical examples of GB∗ -algebras are C ∗ -algebras, pro-C ∗-algebras (i.e., inverse limits of C ∗ -algebras), C ∗ -like locally convex ∗-algebras initiated by A. Inoue and K.-D. Kürsten, the Arens algebra Lω [0, 1] =

∩

Lp [0, 1]

1≤p 0, such that B ⊂ εU . Due to convergence of the sequence (xn ) to x, with respect to  · B , we have that there is n0 ∈ N, such that xn − xB < 1ε , for every n ∈ N, n ≥ n0 . Hence, xn − x ∈ 1ε B, n ≥ n0 ; thus, xn − x ∈ U , for all n ≥ n0 . Therefore, xn → x with respect to τ . The intimate relation between the set A0 , of all bounded elements in A, and the normed algebras A[B], B ∈ B0 is given by the following proposition, for which the next definition is helpful. Definition 2.2.3 A subcollection B1 of B0 is called basic if for every B ∈ B0 , there is some B1 ∈ B1 , such that B ⊂ B1 . Proposition 2.2.4 Let A[τ  algebra and B1 a basic subcollec ] be a locally convex tion of B0 . Then, A0 = A[B] : B ∈ B1 . Proof Let x ∈ A[B], for some B ∈ B1 . Then, for λ > xB we have that λ1 x ∈ B.  Since B 2 ⊂ B, by induction we have B n ⊂ B, for n ∈ N, so that ( λ1 x)n : n ∈   1  N ⊂ B. Therefore, the subset ( λ x)n : n ∈ N of A is bounded, since B is bounded and so x ∈ A0 .  n In the  inverse direction, if x ∈ A0 and λ ∈ C, λ = 0, then the set S ≡ (λx) : n ∈ N is bounded in A[τ ]. It is easy to show that the closure of the absolutely convex hull of S belongs to B0 . Therefore, there is some B ∈ B1 , such that S ⊂ B and so x ∈ A[B].   If for a locally convex algebra A[τ ], all the normed subalgebras A[B], B ∈ B0 are Banach algebras, then A[τ ] is called pseudo-complete . Proposition 2.2.5 If A[τ ] is sequentially complete, then A[τ ] is pseudo-complete. Proof Let B ∈ B0 and (xn )n∈N ⊂ A[B], such that the sequence (xn )n∈N is  · B Cauchy. Since τ is weaker than  · B on A[B], (xn )n∈N is τ -Cauchy and therefore, from the assumption of A[τ ] being sequentially complete, we have that there is an x ∈ A, such that xn → x, with respect to τ . Let ε > 0. Since (xn )n∈N is  · B Cauchy, there is a n0 ∈ N, with xm − xn B < ε4 , for all m, n ≥ n0 . Then, since xm −xn0 → x −xn0 , with respect to τ and B is τ -closed, we have that x −xn0 ∈ ε4 B and thus x ∈ A[B]. So, for m ≥ n0 , xm − x = xm − xn0 + xn0 − x ∈

ε ε ε B + B ⊂ B, 4 4 2

2.2 The Set of Bounded Elements. Radius of Boundedness

13

since B is convex. Therefore, xm − xB < ε, for all m ≥ n0 . Hence, A[B] is sequentially complete, therefore a Banach algebra.   Proposition 2.2.6 If B0 contains a basic subcollection B1 , such that A[B] is a Banach algebra for every B ∈ B1 , then A[τ ] is pseudo-complete. Proof Let B ∈ B0 and (xn )n∈N be a ·B -Cauchy sequence in A[B]. There is some B1 ∈ B1 , such that B ⊂ B1 and hence xB1 ≤ xB , for x ∈ A[B]. Therefore, (xn )n∈N is a ·B1 -Cauchy sequence in A[B1 ], hence there is an element x ∈ A[B1 ], such that xn → x with respect to  · B1 . Following similar arguments to those of the proof of Proposition 2.2.5 we have that x ∈ A[B] and xn → x with respect to  · B . Therefore, A[B] is a Banach algebra and since B is an arbitrary element of B0 , we have that A[τ ] is pseudo-complete.   In the example that follows we are going to show that the converse of Proposition 2.2.5 is not valid. Towards this direction, the notion of convergence in the sense of Mackey will be proven useful (cf. [74, Chapter 3, §5] and/or [131, Chapter IV, 3.]). We recall that in a topological vector space E[τ ], a sequence (xn )n∈N is a Mackey Cauchy sequence if there is some sequence (εn )n∈N of positive numbers, which tends to 0 and a bounded subset B of E, such that ∀ n ∈ N, xn − xm ∈ εn B, for m > n. Furthermore, the sequence (xn )n∈N is Mackey convergent to an element x ∈ E, if there is some sequence (εn )n∈N of positive numbers, which tends to 0 and a bounded subset B of E, such that xn − x ∈ εn B, ∀ n ∈ N. The topological vector space E[τ ] is Mackey complete, if every Mackey Cauchy sequence in E is Mackey convergent. We note that if a locally convex space E[τ ] is metrizable and sequentially complete, then it is also Mackey complete. Indeed, a metrizable locally convex space E[τ ] is bornological (see Definition 4.3.14(2) in Sect. 4.2 and [131, p. 61, 8.1]). But then, E[τ ] becomes a Mackey space (ibid., p. 132, 3.4). This means that the topology τ is the Mackey topology τ (E, E ), E the dual of E[τ ]. Namely, τ (E, E ) is the topology of uniform convergence on the absolutely convex σ (E , E)-compact subsets of E

(cf. Definitition 4.3.11 and (4.3.12) in Sect. 4.3). But, a metrizable and sequentially complete locally convex space is complete, hence E[τ ] is Mackey complete. The example that follows illustrates a locally convex algebra, which is metrizable and pseudo-complete, but not Mackey complete and so a fortiori not sequentially complete. Example 2.2.7 Let A be the algebra of all polynomials in one variable with complex coefficients. We endow A with the topology τ of uniform convergence on compact subsets of the positive real line R+ . Then, clearly A[τ ] is metrizable. Moreover, A0 is the set of all constant functions. Yet, the family (B)A of subsets of A has a

14

2 A Spectral Theory for Locally Convex Algebras

greatest member, say B0 , which consists of all constant functions of absolute value not exceeding 1. Therefore, A[B0 ] is a Banach algebra (see (2.2.1) and (2.2.2)) and so from Proposition 2.2.6 we have that A[τ ] is pseudo-complete. Consider a sequence of polynomials (Pn )n∈N in A, such that Pn (x) =

n  r=0

(−1)r+1

x 2r+1 . (2r + 1)!

(2.2.3)

Then, Pn → sin(x), with respect to τ and the set   B = P ∈ A : |P (x)| ≤ ex , ∀ x ∈ R+ is clearly τ -bounded. Since, |Pn (x) − sin(x)|e−x → 0, uniformly on R+ , we deduce that (Pn )n∈N is a Mackey Cauchy sequence. Nevertheless, (Pn )n∈N is not convergent to any element of A. Hence, A[τ ] is not Mackey complete. Note that this last implication is based on the fact that if a sequence (xn )n∈N in a topological vector space E[τ ] is Mackey convergent to an element, say x ∈ E, then xn → x, with respect to τ . Indeed, since xn → x in the sense of Mackey, there is a sequence of positive real numbers (εn )n∈N , which tends to 0 and a bounded subset V of E, such that xn − x ∈ εn V , for all n ∈ N. Let U be a 0-neighbourhood in E. There is ε > 0, such that V ⊂ εU , hence εn V ⊂ εn εU, n ∈ N. Since εn → 0, 1 there is n0 ∈ N, such that εn < ε+1 , for every n ≥ n0 . Suppose, without loss of generality, that U is also balanced. Then, we have that εn V ⊂ εn εU ⊂ U , for every n ≥ n0 . Therefore, xn − x ∈ U , for all n ≥ n0 , thus xn → x with respect to τ . Proposition 2.2.8 (1) If A[τ ] is a pseudo-complete algebra and B is a closed subalgebra of A, then B[τB ] is also pseudo-complete. (2) If A[τ ] has no identity, then its unitization A1 [τ1 ] is pseudo-complete, if and only if, A[τ ] is pseudo-complete.   (3) If A[τ ] has an identity, say e, then the collection B : B ∈ B0 , e ∈ B is a basic subcollection of B0 . Proof (1) Let (B0 )A , (B0 )B be the corresponding collections of subsets for A[τ ] and B[τ B ] respectively. Given the fact that B is a closed subalgebra of A it is straightforward that (B0 )B ⊂ (B0 )A . Hence, the result follows. (2) If A1 [τ1 ] is pseudo-complete, then from (1) we have that A[τ ] is pseudocomplete. In the inverse direction let A[τ ] be pseudo-complete. For B ∈ (B0 )A1 we will show that A1 [B] is a Banach algebra with respect to the norm  · B . We first note that if (λ, x) ∈ B, then λ ∈ D = {z ∈ C : |z| ≤ 1}: indeed

2.2 The Set of Bounded Elements. Radius of Boundedness

15

since B 2 ⊂ B, by induction we have that (λ, x)n = (λn , xn ) ∈ B, for all n ∈ N, where xn is an element of A. Since B is bounded the fact that (λn , xn ) ∈ B is possible only for those complex numbers λ that belong to D. Consider now B1 = 13 D + 13 B. It is easy to show that B1 ∈ (B0 )A1 . Let B2 = {x ∈ A : (λ, x) ∈ B, for some λ ∈ C}. Then, 13 B2 ⊂ B1 ∩ A. If C denotes the closed absolutely convex hull of 13 B2 , then C ⊂ B1 ∩ A and C ∈ (B0 )A . Hence, by the assumption of pseudo-completeness for A[τ ] we have that A[C] is a Banach algebra. Let A[C]1 [ · 1C ] be the unitization of A[C], with norm (λ, x)1C = |λ| + xC , for λ ∈ C, x ∈ A[C]. If Be ≡ U (A[C]1 ) is the closed unit ball of A[C]1 , with respect to the norm  · 1C , it can easily be shown that 16 B ⊂ Be and therefore A1 [B] ⊂ A[C]1 . From the latter inclusion we have that, on A1 [B], the topology of the norm  · 1C is weaker than that of  · B . Indeed, let us consider a sequence (xn )n∈N ⊂ A1 [B], such that xn → x, with respect to  · B and ε > 0. Then, ∃ m0 ∈ N :

1 1 < ε and n0 ∈ N with xn − xB < , ∀ n ≥ n0 . m0 6m0

Therefore, xn − x ∈

1 1 Be , ∀ n ≥ n0 , so that xn − x1C ≤ < ε, for n ≥ n0 . m0 m0

Hence, xn → x, with respect to the norm  · 1C on A[C]1 . Since τ ≺  · C on A[C] we have that the norm topology  · 1C on A[C]1 is stronger than the product topology induced on A[C]1 by the product topology from A[C] × C. Based on this relation between the two topologies and since B ∈ (B0 )A1 is closed, with respect to the product topology on A1 , we have that B is  · 1C -closed in A[C]1 . Aiming to show that A1 [B] is a Banach algebra, let (xn )n∈N be a  · B -Cauchy sequence in A1 [B]. Since  · 1C ≺  · B on A1 [B], we have that (xn )n∈N is a ·1C -Cauchy sequence in A[C]1 . Since A[C]1 [·1C ] is a Banach algebra, there exists an x ∈ A[C]1 , such that xn → x, with respect to  · 1C . Let ε > 0. Then, ∃ n0 ∈ N : xn − xm B
0, such that C ⊂ εU . Thus, for every b ∈ B, we have that bC ⊂ Lb (εU ) ⊂ εV , that is BC is a bounded subset of A[τ ]. Furthermore, since A is commutative, (BC)2 = B 2 C 2 , hence (BC)2 ⊂ BC. Therefore, the closed absolutely convex hull of BC belongs to B0 . Therefore, by Proposition 2.2.8, there is an F ∈ B, such that BC ⊂ F . Since e ∈ B ∩ C we conclude that B ∪ C ⊂ BC ⊂ F . In case A has no identity, for B, C ∈ B we consider B1 , C1 to be the closed absolutely convex hulls of the sets B ∪ {(λ, 0) : λ ∈ D} and C ∪ {(λ, 0) : λ ∈ D} respectively, where D denotes the closed unit disk in C. Then B1 , C1 ∈ (B0 )A1

2.2 The Set of Bounded Elements. Radius of Boundedness

17

and so, as seen above, there is an F ∈ (B)A1 , such that B1 ∪ C1 ⊂ F . Therefore, B ∪ C ⊂ F ∩ A, where F ∩ A ∈ (B)A .   Corollary 2.2.11 If A[τ ] is a commutative and pseudo-complete locally convex algebra, then A0 is a subalgebra of A. Proof From Proposition 2.2.4 we have that A0 = ∪{A[B] : B ∈ B}. Hence, for x, y ∈ A0 there are B, C ∈ B such that x ∈ A[B] and y ∈ A[C]. By Theorem 2.2.10, there is an F ∈ B such that B ∪ C ⊂ F . Therefore, x + y, xy ∈ A[F ] and thus x + y, xy ∈ A0 .   Remark 2.2.12 Note that if A[τ ] is not commutative, then A0 need not even be a linear subspace. But if A[τ ] is pseudo-complete and x, y ∈ A, such that xy = yx, then taking the closed subalgebra B of A[τ ] generated by x, y, B is a commutative pseudo-complete locally convex algebra (see Proposition 2.2.8(i)), therefore Corollary 2.2.11 implies that xy, x +y ∈ B0 , consequently xy, x +y ∈ A0 too. Corollary 2.2.13 Let A[τ ] be a commutative and pseudo-complete locally convex algebra and B0 = ∪{B : B ∈ B}. Then, B0 is absolutely convex, B02 ⊂ B0 and B0 is absorbent in A0 . Hence, the Minkowski functional  · B0 of B0 is a submultiplicative seminorm on A0 . Proof Let 0 = x ∈ A0 . Then there is B ∈ B such that x ∈ A[B]. Hence, x = λy, 1 λ for some λ ∈ C, y ∈ B. Since B is balanced we have that |λ| x = |λ| y ∈ B ⊂ B0 . 2 Therefore, B0 is absorbent in A0 . The properties that B0 ⊂ B0 and that of B0 being absolutely convex result from the fact that every B ∈ B enjoys the same properties and from Theorem 2.2.10.   For a locally convex algebra A[τ ] a useful tool is the notion of the radius of boundedness for an element x ∈ A. The radius of boundedness β(·) of x is defined by the relation   1 β(x) := inf λ > 0 : ( x)n : n ∈ N is bounded , λ where inf ∅ = +∞. In Sect. 2.3 we are going to see the relation between the spectral radius and the radius of boundedness of an element in a locally convex algebra. In the proposition, which follows some basic properties of β(x) are listed. The proofs of these properties are obvious, so that are omitted. Proposition 2.2.14 Let A[τ ] be a locally convex algebra and x ∈ A. Then, the following hold: (1) β(x) ≥ 0 and β(λx) = |λ|β(x), for λ ∈ C, with the convention that 0 · ∞ = 0. (2) β(x) < +∞,  if and only if, x ∈ A0 .  (3) β(x) = inf λ > 0 : ( λ1 x)n → 0, for n → ∞ .

18

2 A Spectral Theory for Locally Convex Algebras

(4) For x ∈ A0 , if λ ∈ C, such that |λ| > β(x), then ( λ1 x)n → 0, for n → ∞. If 0 < |λ| < β(x), then the set {( λ1 x)n : n ∈ N} is unbounded. Proposition 2.2.15 Let A[τ ] be a locally convex algebra. Then, the  restriction of β  to A0 is the Minkowski functional  · B0 of B0 = B: B∈B .   Proof Let x ∈ A0 and λ > 0, such that the set S = ( λ1 x)n : n ∈ N is bounded. Then the closed absolutely convex hull of S, say B, belongs to B. Thus, B ⊂ B0 and therefore λ1 x ∈ B0 . Hence, xB0 ≤ β(x). For the reverse inequality, let μ > 0 such that x ∈ μB0 . Then, there  is some B ∈ B, such that μ1 x ∈ B. Since B 2 ⊂ B, by induction we have that ( μ1 x)n :    n ∈ N ⊂ B, hence ( μ1 x)n : n ∈ N is bounded. Therefore, 

  1 μ > 0 : x ∈ μB0 ⊂ μ > 0 : ( x)n : n ∈ N is bounded , μ

from which follows that β(x) ≤ xB0 .

 

Based on Corollary 2.2.13 and on Proposition 2.2.15 we have the following result. Corollary 2.2.16 Let A[τ ] be a commutative and pseudo-complete locally convex algebra. Then, the restriction of β to A0 is a submultiplicative seminorm. Corollary 2.2.17 Let A[τ ] be a locally convex algebra and x ∈ A0 . Then, β(x) =  inf xB : B ∈ B, x ∈ A[B] . Proof Let α > 0, such that x ∈ αB0 . Then, there is B ∈ B, such that x ∈ αB. Hence,     inf α > 0 : x ∈ αB0 ≥ inf xB : B ∈ B, x ∈ A[B] and thus from Proposition 2.2.15 we have that   β(x) ≥ inf xB : B ∈ B, x ∈ A[B] . On the other hand,  x ∈ A[B], it is clear that for every ε > 0 :  if B ∈ B with xB + ε ≥ inf α > 0 : x ∈ αB0 . Therefore, inf xB : B ∈ B, x ∈ A[B] ≥ β(x).   The next proposition provides us with two other relations with which the radius of boundedness of an element can be expressed. For a locally convex algebra A[τ ], let A denote the topological dual of A and let τ be a family of seminorms defining

2.2 The Set of Bounded Elements. Radius of Boundedness

19

the locally convex topology τ of A. Consider the following formulae  1 β (x) = sup lim sup|f (x n )| n : f ∈ A and n→∞

1 β

(x) = sup lim sup|p(x n )| n : p ∈ τ , x ∈ A. 

n→∞

Proposition 2.2.18 Let A[τ ] be a locally convex algebra, and x ∈ A. Then, β(x) = β (x) = β

(x). Proof Let x ∈ / A0 . Then, β(x) = +∞ and hence, trivially, |f (x)| ≤ β(x), for every f ∈ A . Consider now f ∈ A and suppose that x ∈ A0 and λ > 0 with λ > β(x). Then, from Proposition 2.2.14(4), we have ( λ1 x)n → 0, for n → ∞. Hence, there 1

is n0 ∈ N, such that |f ( λ1n x n )| ≤ 1, ∀n ≥ n0 . Therefore, lim sup|f (x n )| n ≤ λ, for n→∞

each f ∈ A and λ > β(x). Thus, β (x) ≤ β(x), for all x ∈ A. The same arguments as above hold if in place of the functional f we have a seminorm p ∈ τ . Hence, β

(x) ≤ β(x), x ∈ A. Next we show that β(x) ≤ β (x). For β (x) = +∞ this is trivial, so we suppose that β (x) < +∞. In this case, for λ > 0 with λ > β (x), there is some n0 ∈ N, such that for any 

1 f ∈ A , |f ( x)n | < 1, ∀ n ≥ n0 . λ Therefore, the set {( λ1 x)n : n ∈ N} is weakly bounded and thus τ -bounded by [128, p. 67, Theorem 1]. Hence, λ ≥ β(x) and so β(x) ≤ β (x). The result will be proven once we show that β (x) ≤ β

(x). Indeed, if f ∈ A , then from the continuity of f there is some p ∈ τ and M > 0, such that 1

1

1

|f (x n )| n ≤ M n p(x n ) n , ∀ n ∈ N. Therefore, lim sup|f (x n )| n ≤ lim sup|p(x n )| n ≤ β

(x), 1

n→∞

so β (x) ≤ β

(x).

1

n→∞

 

20

2 A Spectral Theory for Locally Convex Algebras

2.3 Spectrum and Spectral Radius The spectrum of an element x of an arbitrary algebra A, will be denoted by sp(x) or by spA (x), when more than one algebra is involved; more precisely, if A has an identity e sp(x) := {λ ∈ C : λe − x ∈ / GA }. If A has no identity, then spA (x) := spA1 (x). Throughout this book C∗ will denote the one point compactification of C. A partial algebraic structure is defined on C∗ as follows: ∞ + λ = ∞, λ ∈ C; ∞ · λ = ∞, λ ∈ C∗ \{0}; and ∞ = ∞. Now, in case of a locally convex algebra A[τ ], G.R. Allan [4, Definition (3.1)] defined a new spectrum of an element x ∈ A as follows: Definition 2.3.1 Let A[τ ] be a locally convex algebra with an identity e. The spectrum of an element x ∈ A, denoted by σ (x) (or by σA (x), when more than one algebra is involved), is the subset of C∗ defined by     / A0 . σA (x) = λ ∈ C : λe − x has no inverse in A0 ∪ ∞ ⇔ x ∈ In case A has no identity then σA (x) := σA1 (x, 0). Concerning the latter, since from Proposition 2.2.8(2), A[τ ] is pseudo-complete, if and only if, A1 [τ1 ] is pseudo-complete, we can assume without loss of generality that the locally convex algebra A[τ ] has an identity e. The complement of σA (x) in C∗ is denoted by ρA (x) (or simply by ρ(x), if no confusion arises) and is called the resolvent set of x. If A[τ ] is a locally convex ∗-algebra, it is clear that   σA (x ∗ ) = λ : λ ∈ σA (x) , ∀ x ∈ A. Proposition 2.3.2 Let A[τ ] be a locally convex algebra and x ∈ A. If C denotes a maximal commutative subalgebra that contains x, then e ∈ C and σA (x) = σC (x). Also if A[τ ] is pseudo-complete, then C[τC ] is pseudo-complete. Proof Clearly the facts that e ∈ C and that C is closed follow from the assumption of C being a maximal commutative subset of A. Therefore, from Proposition 2.2.8(1) we have that C[τC ] is pseudo-complete if A[τ ] is supposed to be pseudo-complete. If λ ∈ / σA (x) and λ = ∞, then (λe − x)−1 ∈ A0 . Since x ∈ C and C is maximal commutative it is straightforward that (λe − x)−1 ∈ C. So, (λe − x)−1 is a bounded

2.3 Spectrum and Spectral Radius

21

element in C[τC ]. Hence, λ ∈ / σC (x). In case ∞ ∈ / σA (x) then x ∈ A0 . Thus, x is a bounded element in C, hence ∞ ∈ / σC (x). Therefore, σC (x) ⊂ σA (x). The other inclusion is trivial.   Towards the development of a functional calculus for a pseudo-complete locally convex algebra that we are going to describe in Sect. 2.4, the definition of the following function is crucial. Definition 2.3.3 Let A[τ ] be a locally convex algebra and x ∈ A. The resolvent of x is the function R(x) defined by R(x)λ ≡ Rλ (x) = (λe − x)−1 , for all λ, such that the inverse of λe − x is defined. When there is no danger of confusion with respect to which element x the resolvent function is being considered, the symbol Rλ is used instead of Rλ (x). The resolvent of x is thus a function on a subset of C∗ , which takes values in A. We recall that a function f on a subset of C∗ that takes values in a locally convex space (λ0 ) E[τ ] is holomorphic on some open set G of C if the limit lim f (λ)−f exists for λ−λ0 λ→λ0

every λ0 ∈ G. Moreover, f is holomorphic at ∞ if the limit lim f ( λ1 ) exists and the λ→0

function g(λ) =

⎧ ⎨f ( 1 ),

if λ = 0

⎩ lim f ( λ1 ),

if λ = 0

λ

λ→0

is holomorphic in some neighbourhood of 0. In order to obtain some useful results concerning the resolvent Rλ we are going to use the weak topology on A. The reason behind this choice can be seen by the following direct observation: if f is a holomorphic function that takes values in any locally convex algebra A[τ ] and x ∈ A , then the function λ → x (f (λ)) is a complex valued holomorphic function. For any locally convex algebra A[τ ] we have the following result concerning the weak topology σ (A, A ) on A. Lemma 2.3.4 Let A[τ ] be a locally convex algebra. Then, A is also a locally convex algebra with respect to σ (A, A ). Proof It suffices to show that the multiplication in A is separately continuous with respect to σ (A, A ). Let f ∈ A , y ∈ A. Consider the maps fy , f y from A into C, such that fy (x) = f (yx), f y (x) = f (xy), x ∈ A. Then, from the separate continuity of multiplication in A, we have that fy , f y ∈ A , from which the result is easily derived.   Lemma 2.3.5 Let A[τ ] be a locally convex algebra and x ∈ A. Suppose that R(x) is weakly holomorphic at μ (= ∞). Then, R(x) has weak derivatives R (n) (x), n ∈ (n) N, at μ, given by Rμ = (−1)n n!Rμn+1 , n ∈ N.

22

2 A Spectral Theory for Locally Convex Algebras

Proof We note that for λ, μ in the domain of R(x) we have that Rλ − Rμ = (λe − x)−1 − (μe − x)−1

 = (λe − x)−1 (μe − x) − (λe − x) (μe − x)−1 = −(λ − μ)Rλ Rμ . Therefore, based on Lemma 2.3.4 and the assumption for R(x) being weakly holomorphic, hence weakly continuous at μ we have that Rλ − Rμ = −Rλ Rμ → −Rμ2 , with respect to σ (A, A ). λ→μ λ−μ Hence, the result follows for n = 1. Let us suppose that the result holds for all n = 1, 2, . . . , m. Then, (m)

Rλ

(m)

− Rμ λ−μ

= (−1)m m!(λ − μ)−1 (Rλm+1 − Rμm+1 ) = (−1)m m!(λ − μ)−1 (Rλ − Rμ )(Rλm + Rλm−1 Rμ + · · · + Rμm ) = (−1)m+1 m!

m 

Rλr+1 Rμm+1−r .

r=0

We show that R r+1 (x) is weakly continuous at μ, for r = 0, 1, . . . , m. Towards this direction, given any f ∈ A consider the complex valued function ϕ(λ) = f (Rλ ), which is holomorphic at μ by the assumption of the lemma. By the inductive hypothesis and taking into account the continuity of f we have that ϕ (r) (λ) = f (Rλ(r) ) for λ in a neighbourhood of μ and for r = 0, 1, . . . , m. The function ϕ (r) is necessarily continuous at μ for r = 0, 1, . . . , m. Therefore, R r+1 (x) is weakly continuous at μ for r = 0, 1, . . . , m. Hence, we have that (m)

Rλ

(m)

− Rμ λ−μ

→ (−1)m+1 (m + 1)!Rμm+2 , for λ → μ, with respect to σ (A, A ).

Thus, the result of the lemma is established by induction.

 

Lemma 2.3.6 Let A[τ ] be a locally convex algebra and x ∈ A, such that R(x) is holomorphic at ∞ with respect to the weak topology σ (A, A ). If Sλ =

⎧ ⎨R 1 ,

λ = 0,

λ

⎩ lim R 1 , λ→0

λ

λ=0

2.3 Spectrum and Spectral Radius

23

then, in some neighbourhood of 0, Sλ has weak derivatives of all orders given by Sλ(n) = n!x n−1 (e − λx)−(n+1) , for n ∈ N and Sλ = λ(e − λx)−1 .

−1

−1 = λ e − λx . Let Proof For λ = 0 we have that Sλ = R 1 = λ1 e − x λ l be the weak limit of R 1 , as λ → 0 (note that the existence of l is ensured due λ to the assumption of R(x) being weakly holomorphic at ∞). Then, (e − λx)Sλ = Sλ − x(λSλ ) and the right hand side of the previous equation will tend weakly to l, as λ → 0, based on Lemma 2.3.4. On the other hand, we have that (e − λx)Sλ = (e − λx)

1 λ

e−x

−1

= λe → 0 as λ → 0.

−1 Therefore, l = 0; hence, Sλ = λ e − λx holds true for λ = 0 too. Now from the assumption that R(x) is weakly holomorphic at ∞ we have that S(x) is weakly holomorphic in some 0-neighbourhood, say N (similar to R(x), the notation S(x) denotes the function S(x)(λ) ≡ Sλ ). Then, we have that Sλ − Sμ = R 1 − R 1 = − λ

1

μ

λ

−

1 R 1 R 1 , for λ = 0 = μ in N. μ λ μ

Hence, for the weak first derivative of S(x), based on Lemma 2.3.4, we have that −1 2

−2 1 Sλ − Sμ 1 

= 2 Sμ2 = 2 μ e − μx = e − μx . λ→μ λ − μ μ μ

Sμ = lim

Therefore, the formula for the weak derivatives of S(x) in some 0-neighbourhood holds for n = 1. Suppose the result holds for n = 1, . . . , m. Then we have that  Sλ(m) − Sμ(m) m!x m−1  1 1 m+1 m+1 = R − R 1 1 λ−μ λ − μ λm+1 λ μm+1 μ  1  m+1 1 m+1 m!x m−1  1 m+1  R R − + − R = 1 1 1 λ−μ λm+1 μm+1 λ μm+1 λ μ 

m m−1 m m−1

 μ +μ λ + ··· + λ m!x = R m+1 μ−λ 1 m+1 λ−μ μ λm+1 λ   1  1 1 1 −1 m+1 m−1 − R 1 − R m+1 + m!x − 1 m+1 λμ λ μ μ λ μ   1 −(m + 1)! m−1 m+1 1 → x R 1 + m+1 m!x m−1 − 2 λ→μ μm+2 μ μ μ   · − (m + 1)R m+2 1 μ

24

2 A Spectral Theory for Locally Convex Algebras

 x m−1 m+1  1 x m−1 m+1 R R − e = (m + 1)! R xR 1 1 μ μm+2 μ1 μ μ μm+2 μ1

 1 −(m+2) = (m + 1)!x m m+2 R m+2 = (m + 1)!x m e − μx . 1 μ μ = (m + 1)!

 

Therefore, the result follows by induction.

Theorem 2.3.7 Let A[τ ] be a locally convex algebra and x ∈ A. Let R(x) be the resolvent of x. Then, the following hold: (1) If R(x) is weakly holomorphic at μ, μ ∈ C∗ , then μ ∈ ρ(x). (2) If μ ∈ ρ(x), then there is a neighbourhood N of μ and B ∈ B, such that, for every λ ∈ N ∩ ρ(x), Rλ ∈ A[B]. Also, for μ = ∞, R(x) is differentiable at μ relative to ρ(x) in the sense of norm convergence in A[B]. (3) If A[τ ] is pseudo-complete and μ ∈ ρ(x), then the neighbourhood N and the set B ∈ B as in (2) can be chosen, such that N ⊂ ρ(x), Rλ ∈ A[B], for all λ ∈ N and R(x) is holomorphic at μ in the sense of norm convergence in A[B]. Proof (1) Let R(x) be weakly holomorphic at μ = ∞. Then, there exists δ > 0, such that R(x) is weakly holomorphic at λ ∈ C, for |λ−μ| < δ. If f ∈ A , consider the function φ(λ) = f (Rλ ), λ ∈ C. Note that, φ is holomorphic at λ, for every λ, such that |λ − μ| < δ. From Lemma 2.3.5 we have that φ (n) (μ) = f (Rμ(n) ) = (−1)n n!f (Rμn+1 ), n ∈ N. Then, the Taylor expansion of φ about μ is given by φ(λ) =

+∞ 

(−1)k f (Rμk+1 )(λ − μ)k , for |λ − μ| < δ.

k=0 1

Therefore, from Cauchy’s radius-of-convergence formula lim sup|f (Rμn )| n ≤ n

1 δ.

Hence, from Proposition 2.2.18 we have that β(Rμ ) ≤ 1δ . Thus, Rμ ∈ A0 (see Proposition 2.2.14(2)), so μ ∈ ρ(x). In case now R(x) is weakly holomorphic at ∞, we have Sλ =

⎧ ⎨R 1 ,

λ = 0

λ

⎩ lim R 1 , λ→0

λ

λ=0

is weakly holomorphic at 0.

So, for f ∈ A , the function φ(λ) = f (Sλ ) is holomorphic at λ, |λ| < δ, for some δ > 0. From Lemma 2.3.6 we have that φ (n) (0) = f (n!x n−1 ). Therefore, the Taylor expansion of φ about 0 is given by φ(λ) =

+∞  k=0

f (x k−1 )λk , for |λ| < δ.

2.3 Spectrum and Spectral Radius

25 1

Hence, we have that lim sup|f (x n )| n ≤ n

1 δ.

Consequently, by Proposition 2.2.18,

β(x) ≤ 1δ , thus x ∈ A0 and so ∞ ∈ ρ(x). For (2) and (3), we consider two cases below. • Case μ = ∞. Let μ ∈ ρ(x), μ = ∞. Then, Rμ exists and Rμ ∈ A0 . From Proposition 2.2.4 and the comment in , before Proposition 2.2.9, we have that there is B ∈ B, such that Rμ ∈ A[B]. Clearly Rμ B > 0. Let λ ∈ C, with |λ − μ| < Rμ1 B . Moreover, let sn denote the n-th partial sum of the series Rμ − Rμ2 (λ − μ) + Rμ3 (λ − μ)2 − · · ·

(2.3.4)

For m > n we have that sn − sm B ≤ Rμ B

m 

Rμ kB |λ − μ|k → 0, for n, m → +∞.

k=n+1

Hence, (sn )n∈N forms a Cauchy sequence in A[B]. Since τ ≺  · B on A[B], for every seminorm p ∈ τ , there is Cp > 0, such that xB ≤ Cp p(x), x ∈ A[B].   Therefore, for every p ∈ τ and λ, such that λ ∈ N = z ∈ C : |z−μ| < Rμ1 B we have 



p (λe − x)sn − e = p (λ − μ)sn + (μe − x)sn − e

 = p e + (−1)n (λ − μ)n+1 Rμn+1 − e (2.3.5) ≤ Cp |λ − μ|n+1 Rμ n+1 → 0, for n → +∞. B



 So, lim (λe − x)sn = e and similarly lim sn (λe − x) = e, with respect to n

n

τ . Therefore, for λ ∈ N ∩ ρ(x), we have that sn → Rλ , with respect to τ . Furthermore, since (sn )n∈N is a  · B -Cauchy sequence in A[B] there is M > 0, 1 such that sn B ≤ M for all n ∈ N. Hence, M sn ∈ B, for all n ∈ N. Then, 1 1 since M sn → M Rλ , with respect to τ , and given that B is τ -closed, we conclude that Rλ ∈ A[B]. Then, we have that sn → Rλ , with respect to  · B : indeed, by analogous computations as those of (2.3.5), we have that for λ ∈ N, (λe−x)sn − eB = |λ−μ|n+1 Rμ n+1 → 0, as n → ∞. Therefore, (λe−x)(sn −Rλ )B → B 0, for n → ∞. Hence, sn − Rλ B = Rλ (λe − x)(sn − Rλ )B ≤ Rλ B (λe − x)(sn − Rλ )B → 0, as n → ∞. It is clear then from the series in (2.3.4) that R is differentiable at μ relative to ρ(x) in the sense of norm convergence in A[B].

26

2 A Spectral Theory for Locally Convex Algebras

If, in addition, A[τ ] is pseudo-complete, then A[B] is a Banach algebra. Hence, for λ ∈ N, the series in (2.3.4) is necessarily convergent, so Rλ exists and belongs to A[B]. Therefore, λ ∈ ρ(x), so that N ⊂ ρ(x). • Case μ = ∞. Suppose that ∞ ∈ ρ(x). Then, x ∈ A0 , so there is some B ∈ B, such that x ∈ A[B]. Let N = {λ ∈ C∗ : |λ| > xB }. For λ ∈ N ∩ ρ(x), using similar arguments just as in the previous case, we have Rλ ∈ A[B].  that−n Moreover, for λ = ∞, λ ∈ N ∩ ρ(x), we have that Rλ = +∞ λ x n−1 , with n=1 respect to norm convergence in A[B]. If A[τ ] is pseudo-complete, then Rλ exists and belongs to A[B] for λ ∈ N. Moreover, for λ ∈ N, λ = ∞, R is holomorphic at λ and Rλ B ≤

+∞  n=1

λ−n x n−1 B =

1 . |λ| − xB

So, Rλ B → 0, for λ → ∞. It is then easily deduced that R is holomorphic at ∞.   Based on the previous proposition we can now establish the following corollaries. Corollary 2.3.8 Let A[τ ] be a locally convex algebra and x ∈ A. Then, σ (x) = ∅. If A[τ ] is pseudo-complete, then σ (x) is closed. Proof Let us suppose that σ (x) = ∅. Then from Theorem 2.3.7(2) we have that R is holomorphic on the whole complex plane C. Also x ∈ A0 since ∞ ∈ ρ(x). Thus, by using the same argument just as at the end of the proof of Theorem 2.3.7, we have that Rλ → 0, for λ → ∞. Then from Liouville’s theorem, as this is applied in locally convex spaces, we have that Rλ = 0, a contradiction. Moreover, in case A[τ ] is pseudo-complete, from Theorem 2.3.7(3) we obtain that σ (x) is closed.   The following corollary extends to the locally convex case the Gelfand–Mazur theorem. Corollary 2.3.9 Let A[τ ] be a locally convex algebra, such that, for every x ∈ A, there is some nonzero λ ∈ C, such that (λx)n → 0. If moreover A is a division algebra, then A is topologically isomorphic to C. Proof We consider the map φ : C → A[τ ] : λ → λe. It is clear that φ is a topological isomorphism onto its image. Based on the assumption that for every x ∈ A there is a nonzero λ ∈ C, such that (λx)n → 0, we derive that A = A0 . Then, by Corollary 2.3.8, we have that σ (x)∩C = ∅. For λ ∈ σ (x)∩C, the element λe − x has no inverse in A0 = A. Hence, since A is a division algebra, λe = x and thus φ is onto. Therefore, the result follows.   Lemma 2.3.10 Let A[τ ] be a pseudo-complete locally convex algebra, x ∈ A and K a compact subset of ρ(x). Then, there is some B ∈ B, such that Rλ ∈ A[B], for every λ ∈ K.

2.3 Spectrum and Spectral Radius

27

Proof Let C be a maximal commutative subalgebra of A that contains x. Being maximal commutative subalgebra, C is closed and e ∈ C. Since C is a closed subalgebra of A[τ ], C[τ C ] is pseudo-complete by Proposition 2.2.8(1). Furthermore, by Proposition 2.3.2, we have that ρ(x) ≡ ρA (x) = ρC (x), hence for each λ ∈ ρ(x), Rλ ∈ C. Let now λ ∈ K. From Theorem 2.3.7(3) there is a neighbourhood Nλ of λ and a set B ∈ B, such that Rμ ∈ A[B] ∩ C, with μ ∈ Nλ . Since K is compact there are finitely many points λ1 , . . . , λn ∈ K, such that the corresponding neighbourhoods Nλ1 , . . . , Nλn cover K. Let B1 , . . . , Bn be the respective subsets in B, such that Rμ ∈ A[Bi ] ∩ C, for i = 1, . . . , n, where μ ∈ Nλi . Since C is closed and e ∈ C, it is straightforward to show that B ∩ C ∈ BC , for every B ∈ B. Hence, taking into account pseudo-completeness of C, we have that A[B ∩ C] = A[B] ∩ C is a Banach algebra with respect to  · B , for every B ∈ B. Therefore, by using analogous arguments as those in the proof of Theorem 2.2.10, it can be shown that the family {B ∩C : B ∈ B} is outer-directed by inclusion. So, there is some B ∈ B, such that (B1 ∩ C) ∪ (B2 ∩ C) ∪ · · · ∪ (Bn ∩ C) ⊂ B ∩ C. Therefore, Bi ∩ C ⊂ B ∩ C, for i = 1, . . . , n. Hence, Rλ ∈ A[B] ∩ C, for every λ ∈ K.   If A is an arbitrary algebra and x ∈ A, the spectral radius of x will be denoted by r(x), or for distinction by rA (x), with   r(x) := sup |λ| : λ ∈ sp(x) .  In the case of a locally convex algebra A[τ ], considering σ (x) in the place of sp(x), we use the same symbols and the same formula for the spectral radius of an element x in A[τ ], with the convention |∞| = +∞.

The relation between the spectral radius and the radius of boundedness, in a locally convex algebra, is given by the following Theorem 2.3.11 Let A[τ ] be a locally convex algebra and x ∈ A. Then, β(x) ≤ r(x). In case A[τ ] is pseudo-complete, then β(x) = r(x). Proof Let r(x) < +∞, for otherwise the inequality is trivial. Then, ∞ ∈ / σ (x) and so x ∈ A0 . Hence, from Proposition 2.2.4, there is some B ∈ B, such that x ∈ A[B]. As in the proof of Theorem 2.3.7(2) we have that if N = {μ ∈ C : |μ| > xB }, then Rλ = λ−1 e + λ−2 x + · · · , for λ ∈ ρ(x) ∩ N,

28

2 A Spectral Theory for Locally Convex Algebras

with respect to norm convergence in A[B]. Then, for f ∈ A the function φ(λ) = f (Rλ ) is written as follows: φ(λ) = λ−1 f (e) + λ−2 f (x) + · · · , for λ ∈ N ∩ ρ(x).

(2.3.6)

Moreover, by Theorem 2.3.7, φ is holomorphic at λ ∈ C with |λ| > r(x). So, φ has a Laurent expansion in that region, which must coincide with the series in (2.3.6). 1 Thus, we have that lim sup|f (x n )| n ≤ r(x) and so by Proposition 2.2.18, β(x) ≤ n

r(x), x ∈ A. Let us assume now that A[τ ] is pseudo-complete. We show that r(x) ≤ β(x), x ∈ A. Suppose that β(x) < +∞, for otherwise the inequality is trivial. Then, by Proposition 2.2.14(2), x ∈ A0 . Let λ ∈ C, such that |λ| > β(x). By Corollary 2.2.17, there exists B ∈ B, such that x ∈ A[B] and |λ| > xB . Then, as in the proof of Theorem 2.3.7(3), we conclude that Rλ exists in A[B] and so λ ∈ ρ(x). Therefore, for every μ ∈ σ (x), we have that |μ| ≤ β(x), from which the result follows.   In case of a pseudo-complete locally convex algebra A[τ ], the following result provides us with a relation between the spectrum σ (x) of an element x ∈ A0 and the spectra σA[B] (x) = spA[B] (x), for those B ∈ B, such that x ∈ A[B]. Proposition 2.3.12 Let A[τ ] be a pseudo-complete locally convex algebra and x ∈ A0 . Then, the following hold:   (1) σA (x) = σA[B] (x) : B ∈ B, x ∈ A[B] ; (2) rA (x) = inf rA[B] (x) : B ∈ B, x ∈ A[B] . Proof For the proof of both claims let C denote a maximal commutative subalgebra of A containing x. (1) Since A[B] ⊂ A, for every B ∈ B, we have that σA (x) ⊂

  σA[B] (x) : B ∈ B, x ∈ A[B] .

Now let λ(= ∞), such that λ ∈ / σA (x). Then, Rλ ∈ A0 , so there is a B1 ∈ B, such that Rλ ∈ A[B1 ] ∩ C (Proposition 2.2.4). By the outer-directedness of {B ∩ C : B ∈ B} (see Theorem 2.2.10) we have that there exists a B ∈ B, such that Rλ ∈ A[B]∩C and x ∈ A[B]. Hence, λ ∈ / σA[B] (x) and so the inverse inclusion follows. (2) From (1) we have that   rA (x) ≤ inf rA[B] (x) : B ∈ B, x ∈ A[B] .   Consider μ ∈ C, such that μ > rA (x) and let K = λ ∈ C∗ : |λ| ≥ μ . Then, K is a compact subset of ρ(x) and so from Lemma 2.3.10 we have that there is some B ∈ B, such that Rλ ∈ A[B], for all λ ∈ K. By using the same argument as in (1)

2.3 Spectrum and Spectral Radius

29

we can assume that x ∈ A[B]. Then, we have that rA[B] (x) ≤ μ. Hence,   inf rA[B] (x) : B ∈ B, x ∈ A[B] ≤ μ, ∀ μ > rA (x), from which the inverse inequality follows.

 

For a locally convex algebra A[τ ] in which inversion is continuous on the invertible elements of A, the relation between σ (x) and sp(x), x ∈ A (for the respective definitions, see beginning of Sect. 2.3) is described by the following result. In the notation of the following theorem sp(x) stands for the closure of sp(x) in C∗ . Theorem 2.3.13 Let A[τ ] be a locally convex algebra with continuous inversion and let x ∈ A. Then, sp(x) ⊂ σ (x) ⊂ sp(x). Moreover, if A[τ ] is pseudo-complete, then σ (x) = sp(x). Proof It is clear from the respective definitions that sp(x) ⊂ σ (x). For the second inclusion, assume that sp(x) = C∗ , for otherwise the result is trivial and let μ ∈ C, such that μ ∈ / sp(x). Then, there is some δ > 0, such that if λ ∈ C with |λ − μ| < δ, then λ ∈ / sp(x). Under the assumption of continuity of inversion on the invertible elements of A, we have that R(x) is continuous at μ. Then, since Rλ − Rμ = −(λ − μ)Rλ Rμ (see beginning of the proof of Lemma 2.3.5), the function R(x) is differentiable at μ, for every finite point μ in the complement of sp(x) in C∗ . Now if ∞ ∈ / sp(x), then there is M > 0, such that if λ = ∞ and |λ| > M, then λ∈ / sp(x). For these λ’s we then have that Rλ = λ−1 (e − λ−1 x)−1 → 0, as λ → ∞. It follows that R is holomorphic at ∞. Therefore, R is holomorphic on C∗ \ sp(x). Thus, from Theorem 2.3.7(1) we have that σ (x) ⊂ sp(x). In case A[τ ] is pseudo-complete, then by Corollary 2.3.8, σ (x) is closed. Hence, sp(x) ⊂ σ (x), thus the desired equality follows.   Corollary 2.3.14 Let A[τ ] be a pseudo-complete locally convex algebra with continuous inversion. Then, x ∈ A0 , if and only if, sp(x) is bounded. Proof By Theorem 2.3.13, sp(x) = σ (x). Let x ∈ A0 and assume that sp(x) is not bounded. Then, there exists a sequence (λn )n∈N in sp(x), such that λn → ∞. So, ∞ ∈ sp(x) = σ (x), hence x ∈ / A0 , a contradiction. For the reverse implication, assume that sp(x) is bounded. Then, sp(x) is bounded [131, p. 25, 5.1]. So, ∞ ∈ / sp(x) = σ (x) and thus x ∈ A0 .  

30

2 A Spectral Theory for Locally Convex Algebras

2.4 A Functional Calculus In this section we develop a functional calculus for a pseudo-complete locally convex algebra. Let A be a pseudo-complete locally convex algebra and x ∈ A. Let us denote with Fx the set of all complex-valued functions, which are holomorphic on the spectrum σ (x) of x, hence on some neighbourhood of σ (x). By Fx we denote the quotient set of Fx by the equivalence relation ∼, which is given as follows: for f, g ∈ Fx , f ∼ g, if and only if, f equals g on some neighbourhood of σ (x). With the algebraic operations defined pointwise on suitable neighbourhoods of σ (x), it follows that Fx is an algebra. We recall that in case A is a Banach algebra, it is known that for f ∈ Fx and γ a closed rectifiable Jordan curve, such that its interior domain, say D, contains σ (x) and f is holomorphic on D and continuous on D ∪ γ , then the formula f (x) =

1 2πi

 f (λ)Rλ (x)dλ γ

defines a homomorphism f → f (x) of Fx into A, which enjoys certain properties (see [115, Chapter III, §11.6, Theorem 7]). An extension of this result can be established for the case where A[τ ] is a pseudocomplete locally convex algebra. Towards this direction we first show the following Proposition 2.4.1 Let A[τ ] be a pseudo-complete locally convex algebra and x ∈ A with ρ(x) = ∅. Let γ be any rectifiable Jordan arc in ρ(x) ∩ C and f a complexvalued function, which is holomorphic on γ . Then, there  is some B ∈ B, such that Rλ ∈ A[B], for all λ ∈ γ . Furthermore, the integral γ f (λ)Rλ (x)dλ exists, in the sense of norm convergence in A[B] and its value is an element of A[B]. Proof Since the set γ is a compact subset of ρ(x), by Lemma 2.3.10 we have that there is some B ∈ B, such that Rλ ∈ A[B], for all λ ∈ γ . Therefore, the function λ → f (λ)Rλ is a holomorphic function on γ which takes values in the Banach algebra A[B]. Then the result follows from the holomorphic functional calculus for Banach algebras (see [115, Chapter I, § 4.7, Theorem I]).   The following definition is a slight variant of [145, Definition, p. 193]. Definition 2.4.2 A subset D of C∗ is called a Cauchy domain if it fulfills the following conditions: (1) D is open; (2) D has a finite number of components, whose closures are pairwise disjoint; (3) the boundary ∂D of D is a subset of C, which consists of a finite number of closed rectifiable Jordan curves no two of which intersect. The next result can be obtained with the use of Proposition 2.4.1 and by following very similar arguments to those developed in [145, Theorem 4.1] and so its proof is omitted.

2.4 A Functional Calculus

31

Lemma 2.4.3 Let A[τ ] be a pseudo-complete locally convex algebra and x ∈ A, such that ρ(x) = ∅. Then for any f ∈ Fx there is a Cauchy domain D, such that (i) σ (x) ⊂ D; (ii) D  ⊂ (f ), where (f ) denotes the domain of f . Furthermore, the integral ∂D f (λ)Rλ (x)dλ defines an element of A0 , which is independent of the choice of the Cauchy domain D satisfying (i) and (ii). We are now in position to establish a functional calculus for a pseudo-compete locally convex algebra. Theorem 2.4.4 Let A[τ ] be a pseudo-complete locally convex algebra and x ∈ A. Then, there is a homomorphism f → f (x) from Fx into A0 , which is given by the following formulae:  1 (1) if x ∈ A0 , then f (x) = 2πi ∂D f (λ)Rλ (x)dλ, whereD is as in Lemma 2.4.3; 1 (2) if x ∈ / A0 and ρ(x) = ∅, then f (x) = f (∞)e + 2πi ∂D f (λ)Rλ (x)dλ, where D is as before; (3) if ρ(x) = ∅, then Fx contains only constant functions. If f (λ) ≡ c, then f (x) = ce. Furthermore, for all cases, if C is a maximal commutative subalgebra of A, which contains x, then f (x) ∈ A0 ∩ C. If u0 , u1 denote the complex functions u0 (λ) ≡ 1, u1 (λ) ≡ λ, then in case (1), u1 ∈ Fx and u1 (x) = x and in all cases (1)–(3), u0 ∈ Fx and u0 (x) = e. Proof For the proof of (1) and (2) are used standard arguments that follow from similar arguments to those in the proof of [145, Theorem 4.3]. For (3), we have that σ (x) = C∗ . Then, any f ∈ Fx is holomorphic on the whole complex plane and at ∞. Therefore, by Liouville’s theorem as is applied to vector-valued functions (see [115, Chapter I, §3.12]), we have that Fx consists only of constant functions. Moreover, with respect to the fact that f (x) ∈ C ∩ A0 , where C is a maximal commutative subalgebra of A containing x, this follows directly from the fact that e ∈ C and that (λe − x)−1 commutes with all elements of C, for every λ ∈ ∂D. Now if u1 (λ) = λ and x is in A0 , then taking into account Proposition 2.3.12(1) we can choose D, such that its boundary ∂D is a circle of radius greater than xB , where B is some element of B with x ∈ A[B]. Then, for λ ∈ ∂D we have that

−1 (λe − x)−1 = e − λ−1 x λ−1 = λ−1 e + λ−2 x + · · · Hence, u1 (x) =

1 2πi



λ(λe − x)−1 dλ = ∂D

1 2πi





 e + λ−1 x + λ−2 x 2 + · · · dλ = x.

∂D

The statement about u0 follows from similar considerations.

 

32

2 A Spectral Theory for Locally Convex Algebras

2.5 The Carrier Space Consider a commutative and pseudo-complete locally convex algebra A[τ ] with an identity e. Recall that B denotes the collection of subsets of A just as the one described in Proposition 2.2.8(3). It will be convenient for the purposes of this section to denote the elements of B by {Bα : α ∈ }, where  is an index set. For α, β ∈ , we use the notation, α ≤ β if Bα ⊂ Bβ . Then, by Theorem 2.2.10 the index set  is outer-directed by the ordering ≤. To simplify the notation, for each α ∈ , we denote by Aα the Banach algebras A[Bα ]. Let Mα denote the set of all nonzero multiplicative linear functionals on Aα endowed with the weak* topology σ (Mα , Aα ). The respective set of all nonzero multiplicative linear functionals on A0 , denoted by M0 , and endowed with the weak* topology σ (M0 , A0 ) is called the carrier space (or Gelfand space, or maximal ideal space) of A0 . Proposition 2.5.1 There exists a natural homeomorphism, say j , of the carrier space M0 with the projective limit lim Mα , where j (ϕ) = (ϕα )α∈ with ϕα (x) = ← − ϕ(x), x ∈ Aα , ϕ ∈ M0 , α ∈ .

α∈

Proof For α, β ∈  with α ≤ β consider the map παβ : Mβ → Mα , such that παβ (ϕβ ) = ϕβ |Aα . The maps παβ are well-defined, that is ϕβ |Aα is not zero, since e ∈ Aα . It is also clear that παβ are weak ∗-continuous maps, such that παα is the identity map on Mα , for all α ∈  and παβ ◦ πβγ = παγ for α ≤ β ≤ γ in . Therefore, the spaces {Mα : α ∈ } form a projective system. Since for each α ∈ , Aα is a commutative Banach algebra with identity, Mα is a nonempty compact Hausdorff space. Hence, the projective limit lim Mα is a non-empty ← − α∈ compact Hausdorff space (see [35, Chapter I, §9, No. 6, Proposition 8]). The fact that the map j is a homeomorphism is then easily checked.    A direct implication of the previous proposition is that the carrier space M0 of A0 is a non-empty compact Hausdorff space. Lemma 2.5.2 Let A[τ ] be a commutative and pseudo-complete locally convex algebra and x ∈ A0 . Then, x is invertible in A0 , if and only if, ϕ(x) = 0, for every ϕ ∈ M0 . Proof The forward implication is immediate since ϕ(e) = 1, for all ϕ ∈ M0 . For the inverse implication let x ∈ A0 and suppose that ϕ(x) = 0, for every ϕ ∈ M0 . We show that there is α ∈ , such that x ∈ Aα and ϕα (x) = 0, for all ϕα ∈ Mα . From a well-known result in the theory of Banach algebras it will then follow that x is invertible in Aα and thus in A. Let us suppose to the contrary that, for every α ∈ , such that x ∈ Aα , there is ϕα ∈ Mα with ϕα (x) = 0. Let us fix an index δ ∈ , such

2.5 The Carrier Space

33

that x ∈ Aδ and for every α ≥ δ, let Nα denote the set {ϕα ∈ Mα : ϕα (x) = 0}. From the assumption we have made, Nα = ∅, for each α ≥ δ. Also Nα , α ≥ δ, is closed, hence a compact subspace of Mα . Since παβ (Nβ ) ⊂ Nα , for all β ≥ α ≥ δ, we have that {Nα }α≥δ forms a projective system, whose projective limit lim Nα is a ← − α≥δ

non-empty compact Hausdorff space. Let {ψα }α≥δ be an element in limNα . Define ← − α≥δ  {ϕα }α∈ ∈ α∈ Mα as follows ϕα =

 ψα , παβ (ψβ ),

if α ≥ δ otherwise, for some β ≥ α, β ≥ δ.

It is clear that ϕα , α ∈ , is well-defined since if α  δ and β, β ∈ , such that β, β ≥ α, then there exists β

∈  with β

≥ β, β and so παβ (ψβ ) = παβ (πβ β

(ψβ

)) = παβ

(ψβ

) = παβ (πββ

(ψβ

)) = παβ (ψβ ). Moreover, {ϕα }α∈ ∈ lim Mα as can easily be verified. So, based on Proposi← − α∈ tion 2.5.1 there is ϕ ∈ M0 , such that ϕ(x) = ϕδ (x) = ψδ (x) = 0, a contradiction. Thus, the result follows.   Theorem 2.5.3 Let A[τ ] be a commutative and pseudo-complete locally convex algebra and x an element in A0 . Then, σ (x) = {ϕ(x) : ϕ ∈ M0 }. Proof Since x ∈ A0 we have that ∞ ∈ / σ (x). Then, λ ∈ C belongs to σ (x), if and only if, λe − x has no inverse in A0 . From Lemma 2.5.2 this is equivalent to ϕ(λe − x) = 0, for some ϕ ∈ M0 , that is ϕ(x) = λ. Hence, the result follows.   The next result describes the unique extension of any functional ϕ ∈ M0 to a bigger set, namely to Aρ := {x ∈ A : ρ(x) = ∅}. Proposition 2.5.4 Let A[τ ] be a commutative and pseudo-complete locally convex algebra. Then, to each functional ϕ ∈ M0 corresponds a unique C∗ -valued function ϕ on Aρ , such that the following hold: (1) ϕ is an extension of ϕ. The C∗ -valued function ϕ on Aρ is a ‘partial character’ of A, in the following sense: (2) ϕ (λx) = λϕ (x), λ ∈ C, x ∈ Aρ , with the convention that 0 · ∞ = 0; (3) ϕ (x1 + x2 ) = ϕ (x1 ) + ϕ (x2 ), provided that x1 , x2 , x1 + x2 ∈ Aρ and ϕ (x1 ), ϕ (x2 ) are not both ∞; (4) ϕ (x1 x2 ) = ϕ (x1 )ϕ (x2 ), provided that x1 , x 2 , x1 x2 ∈ Aρ and ϕ (x1 ), ϕ (x2 ) are not 0, ∞ in some order.

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2 A Spectral Theory for Locally Convex Algebras

Proof (1) Let x ∈ Aρ and μ ∈ ρ(x), such that μ = ∞. Let y = (μe − x)−1 ∈ A0 and consider ϕ ∈ M0 . If an extension, say ϕ , of ϕ to Aρ satisfying properties (2)–(4) is possible, then provided that ϕ(y) = 0, ϕ must satisfy the relation ϕ (μe − x)ϕ(y) = ϕ(e) = 1 ⇒ ϕ (x) = μ − ϕ(y)−1 .

(2.5.7)

If ϕ(y) = 0, then by (3),(4) we have that ϕ (x) = ∞. So, the equality ϕ (x) = μ − ϕ(y)−1 holds in any case and it is considered as the definition of ϕ . The definition of ϕ is independent from the choice of μ ∈C ∩ ρ(x). Indeed

let μ1 , μ2 ∈ C∩ρ(x). If either of the elements ϕ (μ1 e−x)−1 , ϕ (μ2 e−x)−1 is 0, then the other must be 0 also, as it follows from the relation Rμ1 − Rμ2 = (μ2 −μ1 )Rμ1 Rμ2 (see beginning of the proof of Lemma 2.3.5). If both elements are not 0, then 1 ϕ(Rμ1 − Rμ2 ) 1 −

 =−

ϕ(Rμ1 )ϕ(Rμ2 ) ϕ (μ1 e − x)−1 ϕ (μ2 e − x)−1 =

(μ1 − μ2 )ϕ(Rμ1 Rμ2 ) ϕ(Rμ1 )ϕ(Rμ2 )

= μ1 − μ2 , 1 1 hence μ1 − ϕ((μ e−x) −1 ) = μ2 − ϕ((μ e−x)−1 ) . The previous argumentation 1 2 establishes also the uniqueness of any possible extension of ϕ to Aρ , satisfying conditions (2)–(4). Moreover, ϕ is an extension of ϕ. Indeed if x ∈ A0 , then from ϕ(μe −

x)ϕ (μe − x)−1 = 1 we have that ϕ (μe − x)−1 = 0, therefore



ϕ μ(μe − x)−1 − e 1  = ϕ (x) = μ −

ϕ((μe − x)−1 ) ϕ (μe − x)−1 

ϕ x(μe − x)−1 = ϕ(x). = ϕ((μe − x)−1 )



 (2) Let x ∈ Aρ and μ ∈ ρ(x) ∩ C, such that ϕ (μe − x)−1 = 0. If λ ∈ C, λ = 0, then λμ ∈ ρ(λx), hence ϕ (λx) = λμ −

  1 1 = λ μ − = λϕ (x). ϕ((λμe − λx)−1 ) ϕ((μe − x)−1 )



If λ = 0, then clearly  λϕ (x) = 0 = ϕ (λx). By using similar arguments for the −1 = 0, we derive the result. case ϕ (μe − x)

2.5 The Carrier Space

35

(3) Let x1 , x2 , x1 + x2 ∈ Aρ , such that ϕ (x1 ), ϕ (x2 ) are not both ∞. Consider μ1 ∈ ρ(x1 ) ∩ C, μ2 ∈ ρ(x2 ) ∩ C, λ ∈ ρ(x1 + x2 ) ∩ C. We have that





 ϕ Rλ (x1 + x2 ) (λ − μ1 − μ2 )ϕ Rμ1 (x1 ) ϕ Rμ2 (x2 ) 



+ ϕ Rμ1 (x1 ) + ϕ Rμ2 (x2 )



= ϕ Rλ (x1 + x2 ) ϕ Rμ1 (x1 ) (λ − μ1 − μ2 )e   + μ2 e − x2 + μ1 e − x1 Rμ2 (x2 )

 

= ϕ Rλ (x1 + x2 ) λe − (x1 + x2 ) Rμ1 (x1 )Rμ2 (x2 )



 = ϕ Rμ1 (x1 ) ϕ Rμ2 (x2 ) ,

(2.5.8)

where the last to one equality of A.  due to commutativity 

is derived Therefore, if both ϕ Rμ2 (x2 ) , ϕ Rμ1 (x1 ) are different from 0, then by (2.5.8) we have that ϕ Rλ (x1 + x2 ) = 0 and 1 1 1  = λ − μ1 − μ2 +

+

, ϕ Rλ (x1 + x2 ) ϕ Rμ1 (x1 ) ϕ Rμ2 (x2 )

so that (2.5.7) implies ϕ (x1 + x2 ) = ϕ (x1 ) + ϕ (x2 ).

Also if one of  ϕ(Rμ1 (x1 )) or ϕ(Rμ2 (x2 )) is 0, then from (2.5.8) we have that ϕ Rλ (x1 + x2 ) = 0. Thus, ϕ (x1 + x2 ) = ∞ = ϕ (x1 ) + ϕ (x2 ). (4) Let x1 , x2 , x1 x2 ∈ Aρ . Consider μ1 ∈ ρ(x1 ) ∩ C, μ2 ∈ ρ(x2 ) ∩ C and λ ∈ ρ(x1 x2 ) ∩ C. Then, we have 





ϕ Rλ (x1 x2 ) (λ − μ1 μ2 )ϕ Rμ1 (x1 ) ϕ Rμ2 (x2 ) 



 + μ1 ϕ Rμ1 (x1 ) + μ2 ϕ Rμ2 (x2 ) − 1 



= ϕ Rλ (x1 x2 ) λϕ Rμ1 (x1 )Rμ2 (x2 )   − ϕ (μ1 Rμ1 (x1 ) − e)(μ2 Rμ2 (x2 ) − e) 





= ϕ Rλ (x1 x2 ) λϕ Rμ1 (x1 )Rμ2 (x2 ) − ϕ Rμ1 (x1 )x1 x2 Rμ2 (x2 ) 



= ϕ Rλ (x1 x2 ) ϕ (λe − x1 x2 )Rμ1 (x1 )Rμ2 (x2 ) 



= ϕ Rμ1 (x1 ) ϕ Rμ2 (x2 ) . (2.5.9) 



Suppose that ϕ Rμ1 (x1 ) = 0 and ϕ Rμ2 (x2 ) = 0. The previous two relations, due to the very definition of ϕ , result equivalently in that ϕ (x1 ) = ∞ and

36

2 A Spectral Theory for Locally Convex Algebras



 ϕ (x2 ) = ∞. By (2.5.9), ϕ Rλ (x1 x2 ) = 0 ⇔ ϕ (x1 x2 ) = ∞ . Therefore,

 ϕ Rλ (x1 x2 )x1 x2 1 = ϕ (x1 x2 ) = λ − ϕ(Rλ (x1 x2 )) ϕ(Rλ (x1 x2 )) 

ϕ Rλ (x1 x2 )(μ1 Rμ1 (x1 ) − e)(μ2 Rμ2 (x2 ) − e) = ϕ(Rλ (x1 x2 ))ϕ(Rμ1 (x1 ))ϕ(Rμ2 (x2 ))    1 1 = μ1 − μ2 − ϕ(Rμ1 (x1 )) ϕ(Rμ2 (x2 ))

= ϕ (x1 )ϕ (x2 ). 

Furthermore, it follows from (2.5.9) that ϕ Rλ (x1 x2 ) = 0 if one of the following cases holds: 



(i) ϕ Rμ1 (x1 ) = 0 and ϕ Rμ2 (x2 ) = 0,



 (ii) ϕ Rμ1 (x1 ) = 0 and ϕ Rμ2 (x2 ) = μ12 ,



 (iii) ϕ Rμ1 (x1 ) = μ11 and ϕ Rμ2 (x2 ) = 0.

 For x ∈ Aρ and μ ∈ ρ(x) ∩ C we have that ϕ Rμ (x) = 0, if and only

 if, ϕ (x) = ∞, and ϕ Rμ (x) = μ1 , if and only if, ϕ (x) = 0. Therefore, we conclude that ϕ (x1 x2 ) = ϕ (x1 )ϕ (x2 ) = ∞ in all cases (i)–(iii) above. This completes the proof of (4).   Proposition 2.5.5 Let A[τ ] be a commutative and pseudo-complete locally convex algebra and x ∈ Aρ . Then, σ (x) = {ϕ (x) : ϕ ∈ M0 }. Proof Let μ ∈ ρ(x) ∩ C. Put y ≡ (μe − x)−1 ∈ A0 and z ≡ μe − x. Consider λ ∈ C with λ = 0. Since zy = e, λe − z = −λz(λ−1 e − y). Hence λe − z has an inverse in A, if and only if, λ−1 e − y has an inverse in A. Thus, an easy calculation shows that (λe − z)−1 = λ−1 e + λ−2 (y − λ−1 e)−1 . Therefore, (λe − z)−1 ∈ A0 , if and only if, (λ−1 e − y)−1 ∈ A0 . Now λ = 0 ∈ σ (y) is equivalent to z = y −1 ∈ / A0 , which occurs, if and only if, ∞ ∈ σ (z). So, in any case we have that σ (z) = {λ−1 : λ ∈ σ (y)}. Now, ρ ∈ C ∩ σ (x), if an only if, ρe − x has no inverse in A0 , if and only if, (μ − ρ)e − z has no inverse in A0 , if and only if, (μ − ρ)−1 ∈ σ (y). Note that μ − ρ = 0 since ρ ∈ σ (x) and μ ∈ ρ(x). Hence, there is a λ ∈ σ (y), λ = 0, such that ρ = μ − λ−1 . If ∞ ∈ σ (x), then x ∈ / A0 , so z ∈ / A0 and thus  0 ∈ σ (y). Therefore, in any case we have that σ (x) = μ − λ−1 : λ ∈ σ (y) . Based on

2.5 The Carrier Space

37

Theorem 2.5.3 and the proof of Proposition 2.5.4(1) we conclude that  σ (x) = μ −

  1 : ϕ ∈ M = ϕ (x) : ϕ ∈ M0 . 0 −1 ϕ((μe − x) )  

Proposition 2.5.6 Let A[τ ] be a commutative and pseudo-complete locally convex algebra and x ∈ Aρ . Then, for f ∈ Fx and ϕ ∈ M0 , we have ϕ(f (x)) = f (ϕ (x)). Proof Suppose first that x ∈ / A0 . Then, by Theorem 2.4.4(2) we have that  f (x) = f (∞)e +

f (λ)Rλ (x)dλ, ∂D

for a Cauchy domain D having the properties of Definition 2.4.2. By Proposition 2.4.1 we have that there is some B ∈ B, such that the integral appearing in the definition of f (x) converges with respect to the norm of A[B]. Since A[B] is a commutative Banach algebra, the restriction of ϕ to A[B] is continuous. Therefore, we have that 



 f (λ)ϕ Rλ (x) dλ. ϕ f (x) = f (∞) + ∂D

In case ϕ (x) = ∞, then from the way ϕ

is defined  (see (2.5.7)) we have  ϕ(Rλ ) = 0, for every λ ∈ ρ(x). Thus, in this case, ϕ f (x) = f (∞) = f ϕ (x) . Suppose now that ϕ (x) = ∞. Then, (for Rλ , see Definition 2.3.3) ϕ(Rλ ) = −1

λ − ϕ (x) , hence

 ϕ f (x) = f (∞) +



−1

 f (λ) λ − ϕ (x) dλ = f ϕ (x) , ∂D

where the last equality is derived by Cauchy’s formula for an unbounded domain (see [145, p. 190]). In case x ∈ A0 , then similar considerations to the above (but without the term f (∞)) gives us the result.   The following result provides a spectral mapping theorem for a pseudo-complete locally convex algebra. Proposition 2.5.7 Let A[τ ] be a pseudo-complete locally convex algebra and x ∈

 A. Then, for any f ∈ Fx , σ f (x) = f σ (x) . Proof Let us suppose first that x ∈ Aρ . If C is a maximal commutative subalgebra containing x, then by Theorem 2.4.4, f (x) ∈ A0 ∩ C. Let (M0 )C denote the carrier space of A0 ∩ C. Then, by Proposition 2.3.2, Theorem 2.5.3, Proposition 2.5.5 and

38

2 A Spectral Theory for Locally Convex Algebras

Proposition 2.5.6 we have the following:



 

  σA f (x) = σC f (x) = ϕ f (x) : ϕ ∈ (M0 )C 

  

= f ϕ (x) : ϕ ∈ (M0 )C = f σC (x)

 = f σA (x) . In case x ∈ / Aρ , that is ρ(x) = ∅, then by Theorem 2.4.4(3), Fx contains only  constant functions. So, if f (λ) ≡ c ∈ F , then it is clear that σ f (x) = σ (ce) = x  {c} = f σ (x) .   Notes All the results presented in this chapter are due to G.R. Allan and can be found in [4].

Chapter 3

Generalized B*-Algebras: Functional Representation Theory

Having the background we need from Chap. 2, we introduce in the present chapter GB∗ -algebras (abbreviation for generalized B ∗ -algebras (Sect. 3.3)), originated by G.R Allan, in 1967. These algebras generalize the celebrated C ∗ -algebras and they are often met in Analysis. A typical example of a GB∗ -algebra is the Arens algebra Lω [0, 1]; for this and other examples, see 3.3.16. In this chapter we are mainly concerned with commutative GB∗ -algebras. In Sect. 3.4, Theorem 3.4.9 is proved, which is an “algebraic” commutative Gelfand– Naimark type theorem for GB∗ -algebras; namely, it is shown that every commutative  of GB∗ -algebra A[τ ] with identity is algebraically ∗-isomorphic to the ∗-algebra A ∗ all continuous C (:= C ∪ {∞})-valued functions on the Gelfand space M0 of the commutative C ∗ -subalgebra A0 = A[B0 ] of A[τ ]. In the final Sect. 3.5, we exhibit the definition and basic properties of C ∗ -like locally convex ∗-algebras, introduced by A. Inoue and K.-D. Kürsten, in 2002. In Theorem 3.5.3, we prove that these topological ∗-algebras are GB∗ -algebras. Moreover, we compare in the case of a GB∗ -algebra the C ∗ -algebra A[B0 ] with the Banach ∗-algebra D(p ); see (3.3.9), (3.5.24), as well as Proposition 3.5.6 and the discussion that follows. The first two Sects. 3.1, 3.2 of this chapter constitute a preparatory stage for the third Sect. 3.3. In particular, Sect. 3.2 shows how G.R. Allan was led to the definition of GB∗ -algebras.

3.1 Hermitian and Symmetric Locally Convex *-Algebras Section 3.1, together with Chap. 2, provide us with all ingredients needed for the definition and the study of the main theory of GB∗ -algebras that are discussed in this book.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Fragoulopoulou et al., Generalized B*-Algebras and Applications, Lecture Notes in Mathematics 2298, https://doi.org/10.1007/978-3-030-96433-7_3

39

40

3 Generalized B*-Algebras: Functional Representation Theory

As we shall see in Sect. 3.3, a GB∗ -algebra is by definition a locally convex ∗-algebra with some extra conditions, among them being symmetry, a property closely related with hermiticity. In the present section, some general comments relating the preceding (algebraic) concepts are presented and then we specialize in the properties and relationship of these concepts within the context of pseudocomplete (see discussion before Proposition 2.2.5) locally convex ∗-algebras. We start with some standard algebraic notions that subsequently are combined with topological ones. If A is an algebra, we shall call involution, a map ∗ : A → A : x → x ∗ , with the following properties: (λx + y)∗ = λx ∗ + y ∗ , (xy)∗ = y ∗ x ∗ , (x ∗ )∗ = x, for all x, y ∈ A and λ, ∈ C. An algebra A endowed with an involution ∗ will be called a ∗-algebra. A subalgebra B of A invariant under involution is called a ∗-subalgebra. An element x ∈ A is called self-adjoint, respectively normal, if x ∗ = x, respectively x ∗ x = xx ∗ . The notation H (A), N(A) will stand for the sets of self-adjoint, respectively normal elements of A. It is clear that H (A) is a real vector subspace of A, such that A = H (A) ⊕ iH (A),

(3.1.1)

where i is the imaginary unit. If A is a ∗-algebra without identity, its unitization A1 (see beginning of Sect. 2.1) becomes a ∗-algebra, by defining involution as follows (x, λ)∗ := (x ∗ , λ), ∀ x ∈ A and λ ∈ C.

(3.1.2)

If A[τ ] is a locally convex algebra (see Definition 2.1.1) and ∗ an involution on A, this always will be considered continuous. In this case, the real vector space H (A) is closed. Definition 3.1.1 A topological algebra A[τ ] (ibid.) endowed with a continuous involution will be called a topological ∗-algebra. When the underlying topological vector space of a topological ∗-algebra A[τ ] is a locally convex space, then we shall speak of a locally convex ∗-algebra. A metrizable and complete locally convex ∗-algebra is called a Fréchet ∗-algebra. A normed (resp. Banach) algebra, with continuous involution, is said to be a normed ∗-(resp. Banach ∗-) algebra. An algebra norm on a normed ∗-algebra, which preserves involution is called an m∗ norm. It is known that if E, F, G are topological vector spaces, where E, F are metrizable with E also barrelled and G locally convex, then every separately continuous bilinear map from E × F into G is continuous [74, p. 357, Theorem 1]. Hence, every Fréchet locally convex (∗-)algebra A[τ ], being barrelled, has

3.1 Hermitian and Symmetric Locally Convex *-Algebras

41

continuous multiplication. In this case, one easily shows that the topology τ is defined by a sequence ≡ {pn }n∈N of (∗−)seminorms, such that pn (x) ≤ pn+1 (x) and pn (xy) ≤ pn+1 (x)pn+1 (y),

(3.1.3)

for all x, y ∈ A, n ∈ N. For completeness sake, we give a short proof of the second inequality in (3.1.3). Let p1 ∈ . By the joint continuity of multiplication in A, there exist n1 , m1 ∈ N, such that p1 (xy) ≤ pn1 (x)pm1 (y), ∀ x, y ∈ A. Hence, we have p1 (xy) ≤ p2 (x)p2 (y), ∀ x, y ∈ A, with p1 := p1 and p2 := pmax{n1 ,m1 } . Applying the preceding argument to p2 , we obtain p2 (xy) ≤ p3 (x)p3 (y), ∀ x, y ∈ A, with p3 := pmax{n2 ,m2 } . Continuing in this way, we find sequences of natural numbers (nk )k∈N , (mk )k∈N , such that



pk (xy) ≤ pk+1 (x)pk+1 (y), ∀ x, y ∈ A, with pk+1 := pmax{nk , mk } .

Notice now that the family of seminorms ≡ {pn }, n ∈ N, satisfies (3.1.3) and besides, equivalently defines the given topology τ of A. This completes the proof of (3.1.3). Furthermore, observe that in a Banach algebra A[·] with continuous involution ∗, we can always suppose, without loss of generality, that the involution is isometric. Indeed, the function x := max{x, x ∗ }, x ∈ A, is a ∗-preserving norm on A, equivalent to  · , making A a Banach algebra with isometric involution. If A[τ ] is a topological ∗-algebra without identity, its unitization A1 [τ1 ] (see discussion after Definition 2.1.1) endowed with the involution given by (3.1.2) is also a topological ∗-algebra. By a topological (∗-)isomorphism between two topological (∗-)algebras, we mean an algebraic (∗-)isomorphism (i.e., a bijective (∗-)homomorphism), which is a homeomorphism. When we use the symbol “∼ =” between two topological (∗-) algebras, this will always mean a topological (∗-)isomorphism. By a topological (∗-)embedding, we mean an algebraic (∗-)monomorphism, which is bicontinuous on its image.

42

3 Generalized B*-Algebras: Functional Representation Theory

If A is a ∗-algebra and p is a seminorm (resp. norm) on A, we say that p has the C ∗ property or equivalently p is a C ∗ -seminorm (resp. a C ∗ -norm), if p(x ∗ x) = p(x)2 , for all x ∈ A. Z. Sebestyén proved, in 1979, that every C ∗ -seminorm on a ∗-algebra is automatically ∗-preserving and submultiplicative (see, for instance, [52, p. 167, §38]). In this regard, we now set the following Definition 3.1.2 A locally convex algebra A[τ ] with involution, whose topology τ is defined by an upwards directed family of C ∗ -seminorms is called a C ∗ convex algebra; its involution is automatically continuous, as follows from the above comments. A complete C ∗ -convex algebra, is called a pro-C ∗ -algebra [124] (this is what Apostol [8] called a b∗ -algebra, Inoue [75] a locally C ∗ -algebra and Schmüdgen [132] an LMC∗ -algebra). A metrizable pro-C ∗ -algebra is said to be a σ C ∗ -algebra [124]. A ∗-algebra endowed with a C ∗ -norm is called a pre C ∗ -algebra. A complete pre C ∗ -algebra is said to be a C ∗ -algebra. Clearly, every C ∗ -algebra is a pro-C ∗ -algebra. It is obvious from the preceding comments that every pro-C ∗-algebra is a complete m∗ -convex algebra, in the sense that all C ∗ -seminorms defining its topology are m∗ -seminorms, that is ∗-preserving m-seminorms (see discussion before Definition 3.1.2). Let A be a ∗-algebra. The involution of A is said to be hermitian, whenever the spectrum sp(h) (see beginning of Sect. 2.3) of any element h ∈ H (A) is a subset of the real line. A ∗-algebra with a hermitian involution is called a hermitian algebra; cf., e.g., [52, p. 128, (32.2)]. Furthermore, a ∗-algebra A is called symmetric if for every x ∈ A, the element x ∗ x is quasi-invertible. Every symmetric algebra is hermitian [52, p. 129, (32.4) Corollary]; the converse is true for every Banach algebra with involution and this result is known as the Shirali–Ford theorem; see [52, p. 136, (33.2) Theorem] and [60, p. 297, Theorem 22.23]; in the latter, the reader will find a proof based on Pták’s theory [126] for hermitian algebras. Thus, symmetry and hermiticity coincide on Banach algebras with involution. In the non-normed case, one has that every pro-C ∗ -algebra (and a fortiori every C ∗ -algebra) is symmetric [60, p. 268, (21.6)]. An analogue of the Shirali–Ford theorem, in the aforementioned setting is valid for every hermitian spectral complete m-convex algebra (see [60, Theorem 22.27]). Note that an algebra A is called spectral (Palmer, [120]), if it can be equipped with an m-seminorm p, such that rA (x) ≤ p(x), for all x ∈ A.  An element x ∈ H (A) with sp(x) ⊆ [0, +∞), respectively sp(x) ⊆ (0, +∞), is called positive, respectively strictly positive and we write x ≥ 0, respectively x > 0. The set of all positive elements of A will be denoted by A+ .

3.1 Hermitian and Symmetric Locally Convex *-Algebras

43

In this regard, a geometric characterization of symmetry, reads as follows: a ∗algebra A is symmetric, if and only if, x ∗ x ≥ 0, for every x ∈ A (see [52, (32.5) Proposition]). In the case of a locally convex ∗-algebra A[τ ], the definition of “hermiticity” and “symmetry” is further specialized by using the notion of (Allan-)bounded elements (see Definitions 2.2.1 and 2.3.1). Definition 3.1.3 (Allan) The involution on a locally convex ∗-algebra A[τ ] is said to be hermitian if σ (h) is real, for every h ∈ H (A); note that ∞ is counted real, in this case. When a locally convex ∗-algebra A[τ ] has a hermitian involution, it will be called hermitian. Recall that given a locally convex algebra A[τ ], A0 denotes the set of all bounded elements of A[τ ]; see Definition 2.2.1 and the comments that follow. It is evident that if A[τ ] is a locally convex ∗-algebra and x an element of A, then x ∈ A0 , if and only if, x ∗ ∈ A0 . Lemma 3.1.4 Let A[τ ] be a hermitian pseudo-complete locally convex ∗-algebra with identity e. Then, for every h ∈ H (A), the following hold: (i) the element e + h2 has a bounded inverse; (ii) the element h(e + h2 )−1 is bounded. Proof (i) Since the involution of A is hermitian, we have that i ∈ ρ(h) and i ∈ ρ(−h), for every h ∈ H (A). Therefore, the elements (ie − h)−1 and (ie + h)−1 exist and belong to A0 . Since moreover, they commute and A[τ ] is pseudo-complete, their product belongs also to A0 (see e.g, Corollary 2.2.11). Hence, we obtain that the element −(ie − h)(ie + h) = e + h2 ,

(3.1.4)

has an inverse that belongs to A0 , for every h ∈ H (A). (ii) Let now u ≡ h(e + h2 )−1 , h ∈ H (A). Then, u2 = h2 (e + h2 )−2 = (e + h2 )−1 − (e + h2 )−2 , ∀ h ∈ H (A), where since (e + h2 )−1 and (e + h2 )−2 commute and A[τ ] is pseudo-complete, we conclude that u2 ∈ A0 (see, for instance, Corollary 2.2.11). This implies that the set   S = (λu)2n : n ∈ N , for some λ ∈ C\{0},   is bounded. It is now clear that (λu)n : n ∈ N ⊂ S ∪ (λu)S, therefore it is bounded; i.e., u ∈ A0 and this completes the proof.   The next Proposition 3.1.5 provides a characterization of hermiticity.

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3 Generalized B*-Algebras: Functional Representation Theory

Proposition 3.1.5 Let A[τ ] be a pseudo-complete locally convex ∗-algebra with identity e. The following are equivalent: (i) A[τ ] is hermitian; (ii) the element e + h2 has a bounded inverse, for every h ∈ H (A). Proof (i) ⇒ (ii) follows from Lemma 3.1.4(i). (ii) ⇒ (i) By the assumption (ii) and (3.1.4), we conclude that the element ie − h, h ∈ H (A) is invertible with inverse given by (ie − h)−1 = −(ie + h)(e + h2 )−1 = −h(e + h2 )−1 − i(e + h2 )−1 , ∀ h ∈ H (A). By Lemma 3.1.4, this implies that (ie − h)−1 ∈ A0 , for every h ∈ H (A), which in its turn gives that i ∈ / σ (h), for all h ∈ H (A). Suppose now that α, β ∈ R, with β = 0. Then, β −1 (h − αe) ∈ H (A). Therefore, i∈ / σ (β −1 (h−αe)), which equivalently means that α+iβ ∈ / σ (h), for any α, β ∈ R, with β = 0, and h ∈ H (A). This completes the proof.   Definition 3.1.6 (Allan) A locally convex ∗-algebra A[τ ] is called symmetric, if for every x ∈ A, the element x ∗ x is quasi-invertible with bounded quasi-inverse, which in the case of an identity element e, equivalently means that, for every x ∈ A, the element (e + x ∗ x)−1 exists and belongs to A0 . In other words, every symmetric locally convex ∗-algebra A[τ ] is “algebraically” q symmetric, so that for every x ∈ A, x ∗ x ∈ GA and moreover (x ∗ x)◦ ∈ A0 . An immediate consequence of Definition 3.1.6 and Proposition 3.1.5 is the following Corollary 3.1.7 Every symmetric pseudo-complete locally convex ∗-algebra is hermitian. The following Proposition 3.1.8 says that a locally convex ∗-algebra is symmetric, whenever it is “algebraically” symmetric and has a continuous inversion. Proposition 3.1.8 Let A[τ ] be a locally convex ∗-algebra with identity e and continuous inversion (take, for instance an m-convex algebra). Suppose that for every x ∈ A, (e + x ∗ x)−1 exists in A. Then A[τ ] is symmetric. Proof By the discussion before Definition 3.1.3, we have that since A is “algebraically” symmetric, then x ∗ x ≥ 0, for every x ∈ A. This yields that

 sp(e + x ∗ x) ⊂ [1, ∞) and sp (e + x ∗ x)−1 ⊂ (0, 1], ∀ x ∈ A. Now, since A[τ ] has continuous inversion, by Theorem 2.3.13, we conclude that  



σ (e + x ∗ x)−1 ⊂ sp (e + x ∗ x)−1 ⊂ [0, 1], ∀ x ∈ A,

3.2 Some Results on C*-Algebras

45



where the closure of sp (e + x ∗ x)−1 is taken in C∗ . It is now clear that ∞ ∈ / 

∗ −1 σ (e + x x) , for all x ∈ A. This, by definition of the spectrum in a locally convex algebra (see Definition 2.3.1), equivalently means that (e + x ∗ x)−1 ∈ A0 , for all x ∈ A, so that the proof is complete.  

3.2 Some Results on C*-Algebras G.R. Allan noticed in [3], that studying certain locally convex algebras, the following problem cropped up: to express the C ∗ -condition in a given normed ∗-algebra using not its norm, but its properties as a locally convex ∗-algebra. A solution to this problem was given in the previous reference and we present it in this section. An application of these lines of thought were essentially applied for the development of the theory of GB∗ -algebras (see [3, 5], as well as Chap. 2 and Sect. 3.3). Let A[ · ] be a normed algebra with an involution ∗. For convenience, we assume throughout this section that our algebras have an identity element, although all the results can be easily proved without this assumption. Denote by B∗A (see also Definition 3.3.1, in Sect. 3.3) the collection of all subsets B in A[ · ], such that: (1) B is absolutely convex; (2) e ∈ B, B 2 ⊂ B and B ∗ = B, where B ∗ := {x ∗ : x ∈ B}; (3) B is bounded and closed.  We shall consider B∗A endowed with the partial ordering given by inclusion. We are going to give a characterization of a pre C ∗ -algebra, respectively C ∗ algebra (see Theorems 3.2.9 and 3.2.10) through the collection B∗A . This will be done by using the given topological algebra as a locally convex algebra, rather than as a normed algebra. For the proof of the mentioned results we need a series of lemmas that we first axhibit. Lemma 3.2.1 Let A[ · ] be a normed ∗-algebra with identity e and B a ∗subalgebra of A containing e. Suppose that, for every x ∈ B, the element (e + x ∗ x)−1 exists in A. Then, the following hold: (i) for every h ∈ H (B), spA (h) ⊂ R (i.e., the spectrum of every self-adjoint element in B is real); (ii) for every x ∈ B, spA (x ∗ x) ⊂ [0, ∞) (i.e., the spectrum of any element x ∗ x, x ∈ B, is real and non-negative).

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3 Generalized B*-Algebras: Functional Representation Theory

Proof (i) Let h ∈ H (B). Suppose that α + iβ ∈ spA (h), where α, β ∈ R, with β = 0.  Then, considering the polynomial f (t) = β −1 (α 2 + β 2 )−1 αt 2 + (β 2 − α 2 )t , t ∈ C, we notice that f (α + iβ) = i. Applying the spectral theorem

mapping  (see, e.g., [127, Theorem (1.6.10)]), we obtain that i ∈ spA f (h) ; by the same

 reason i 2 ∈ spA f (h)2 , where f (h) ∈ H (B). Hence, e−i 2 f (h)2 = e+f (h)2 does not have an inverse in A, which contradicts our hypothesis. Consequently, β = 0, therefore the spectrum of h is real. 1 (ii) Let λ ∈ R, with λ > 0. Consider the element z := λ 2 x, x ∈ B. Then, z ∈ B and z∗ z = λx ∗ x ∈ H (B). Therefore by (i) z∗ z has a real spectrum. Moreover, e − (−λ)x ∗ x = e + z∗ z is invertible in A by our hypothesis. Hence, −λ ∈ / spA (x ∗ x), which completes the proof of the assertion (ii).   If the involution in the preceding normed algebra A is supposed to be continuous, then without loss of generality we may suppose that x ∗  = x, for every x ∈ A. Thus, the closed unit ball U (A) := {x ∈ A : x ≤ 1} in A is contained in B∗A . In particular, if B0 is the greatest member in B∗A , one obtains U (A) ⊂ B0 ⊂ εU (A), for some ε > 0, which implies that the gauge function (see discussion after Definition 2.2.2)  · B0 is a norm on A equivalent to the given one. This allows us to suppose, without loss of generality, that B0 is the closed unit ball of A.  the completion of A[ · ]. Then, clearly A  is a Banach ∗-algebra. Denote by A Lemma 3.2.2 Let A[ · ] be a normed algebra with identity e and continuous involution. Suppose also that the collection B∗A has a greatest member, say B0 .  Then, h = rA(h), for every h ∈ H (A). Proof First, we suppose that h ∈ H (A). Assume that h ∈ B0 . Then, clearly h2 ∈ B0 . Conversely, let h2 ∈ B0 . Then, choosing ε ∈ R with ε > 1, such that h ∈ εB0 , we obtain h2n ∈ B0n ⊂ B0 ⊂ εB0 and h2n+1 ∈ εB0 B0 ⊂ εB0 , n ∈ N. It is now evident that if C = {e, hn , n ∈ N}, then C is a bounded set and C 2 ⊂ C; therefore the closed absolutely convex hull of C belongs to B∗A and C ⊂ B0 , which yields h ∈ B0 . So we have proved h ∈ B0 ⇔ h2 ∈ B0 . This implies h2  = h2 (see the discussion before Lemma 3.2.2).  Then, there is a sequence (xn )n∈N in A such that Suppose now that h ∈ H (A). xn → h. Put hn = 12 (xn + xn∗ ). Clearly, h∗n = hn , for each n ∈ N and hn → h.

3.2 Some Results on C*-Algebras

47

Thus, h2  = lim h2n  = lim hn 2 = h2 . n

n

Inductively, we obtain  h2  = h2 , n ∈ N, ∀ h ∈ H (A). n

n

Applying now the spectral radius formula, valid in a Banach algebra, we have completed the proof.   Lemma 3.2.3 Let A[ · ] be a normed algebra with identity e and continuous involution. Suppose that the following condition holds: for each x ∈ A, there exist sequences (un )n∈N , (vn )n∈N in A, such that un (e + x ∗ x) → e ← (e + x ∗ x)vn .

(3.2.5)

 Then, for every x ∈ A, the element (e + x ∗ x)−1 exists in A. Proof Put y = e + x ∗ x, x ∈ A. Then, by (3.2.5), there exists a sequence (un )n∈N in A, such that un y → e. So, there is m ∈ N, with um y − e < 1. This implies that  therefore there is u ∈ A,  such that (uum )y = e. Similarly, um y is invertible in A;  there exists l ∈ N and v ∈ A, with y(vl v) = e. It is now clear that y is invertible in  and this completes the proof. A   Lemma 3.2.4 Let A[ · ] be as in Lemma 3.2.3. Suppose also that the collection  is a symmetric Banach algebra. B∗A has a greatest member. Then, A  the element e + x ∗ x is invertible in A.  Proof We must show that, for every x ∈ A,  there is a sequence (xn )n∈N in A, such that xn → x. We Taking an arbitrary x ∈ A, set y = e + x ∗ x and yn = e + xn∗ xn , n ∈ N. Clearly, the elements y, yn , n ∈ N  for all n ∈ N. Thus, applying are self-adjoint and by Lemma 3.2.3, yn−1 ∈ A,  in the place of A and A in the place of B, we have that Lemma 3.2.1, with A spA(xn∗ xn ) is real and non-negative, for all n ∈ N. Hence, spA(yn ) ⊂ [1, ∞), which yields spA(yn−1 ) ⊂ (0, 1], ∀ n ∈ N. Lemma 3.2.2 now gives us that yn−1  = rA(yn−1 ) ≤ 1, ∀ n ∈ N. Furthermore, −1 −1 −1 yn−1 −ym  = ym (ym −yn )yn−1  ≤ ym ym −yn yn−1  ≤ ym −yn , n, m ∈ N.

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But, yn → y, so that (yn−1 )n∈N is a Cauchy sequence, therefore it converges to an  It follows that yz = e = zy and this completes the proof. element z ∈ A.   Lemma 3.2.5 Let A[ · ] be a normed algebra with identity e and continuous involution. Suppose that the following conditions hold: (i) the collection B∗A has a greatest member; (ii) for each x ∈ A, there exist sequences (un )n∈N , (vn )n∈N in A, such that un (e + x ∗ x) → e ← (e + x ∗ x)vn .  becomes a C ∗ -algebra under Then, any maximal commutative ∗-subalgebra C of A ∗  to C. a C -norm equivalent to the norm  · C , the restriction of the norm of A  Then, C is a closed Proof Let C be a maximal commutative ∗-subalgebra of A.  ∗-subalgebra of A with spC (x) = spA(x), for every x ∈ C (see [127, Theorem  is symmetric by Lemma 3.2.4, C is symmetric too (4.1.3)]). Moreover, since A (ibid., Corollary (4.7.7)). Let r(x) ≡ rC (x) = rA(x), x ∈ C. Take an arbitrary x ∈ C. Then, x = h + ik with h, k ∈ H (C). Clearly, hk = kh. Thus, from Lemma 3.2.2 and properties of the spectral radius, we obtain r(x) ≤ xC ≤ hC + kC = rC (h) + rC (k) ≤ 2r(x).

(3.2.6)

To obtain the last inequality, just calculate the spectral radii of h and k using the expressions: h = 12 (x + x ∗ ), k = 2i1 (x − x ∗ ) and standard properties of the spectral radius. Then, (3.2.6) shows that r is a norm equivalent to ·C . But, C is a symmetric Banach algebra, hence hermitian [127, Theorem (4.7.6)], therefore r fulfills the C ∗ property (ibid., Lemma (4.2.1)). Thus, C[r] is a C ∗ -algebra and this completes the proof.   Definition 3.2.6 Given an algebra A, we call the intersection of the kernels of all irreducible representations of A, the Jacobson radical of A and we denote it by JA (see [32, p. 124, Definition 13]). For other expressions of JA we refer to (ibid., p. 124, Proposition 14 and p. 125, Proposition 16). When JA = {0}, A is called semisimple; see, for instance [127, Definition (2.3.1)].  is semisimple. Lemma 3.2.7 Let A[ · ] be as in Lemma 3.2.5. Then, A  is trivial. Let x ∈ J  be arbitrary. Proof We show that the Jacobson radical JA of A A  Then, x = h + ik, for some h, k ∈ H (A). Since, JA is a ∗-ideal and h, k are linear combinations of x, x ∗ , we conclude that h, k ∈ JA. But, every element in the Jacobson radical has trivial spectral radius (cf., e.g., [60, Proposition 4.24(1)] and/or [52, Theorem (B.5.17)(c)]), therefore rA(h) = 0 = rA(k). Now, Lemma 3.2.2 gives h = 0 = k; hence x = 0 and this completes the proof.  

3.2 Some Results on C*-Algebras

49

Definition 3.2.8 ([126]) Let A[ · ] be a Banach algebra with involution ∗ and rA its spectral radius. Then, the function pA (x) := rA (x ∗ x)1/2, x ∈ A, is called the Pták function and plays an important role in the theory of hermitian algebras developed by V. Pták [126], in 1972. Theorem 3.2.9 Let A[ · ] be a normed algebra with identity e and continuous involution. Then, A is a pre C ∗ -algebra under a C ∗ -norm equivalent to the given one, if and only if, the following conditions hold: (i) the collection B∗A has a greatest member; (ii) for each x ∈ A, there exist sequences (un )n∈N , (vn )n∈N in A, such that un (e + x ∗ x) → e ← (e + x ∗ x)vn . Proof “Only if” part. Suppose that A[ · ] is a pre C ∗ -algebra. Then, by the properties of the closed unit ball U (A) of A[ · ], we have that U (A) is a greatest  of A[ · ] is a C ∗ -algebra, member in B∗A . On the other hand, the completion A therefore symmetric (see [127, Theorem (4.8.9)]), consequently (e + x ∗ x)−1 exists  for every x ∈ A. Hence, for every x ∈ A, there is a sequence (un )n∈N in A, in A, such that un → (e + x ∗ x)−1 , which yields that un (e + x ∗ x) → e ← (e + x ∗ x)vn , with vn = u∗n , for every n ∈ N.  of A[ · ] is a semisimple symmetric Banach “If ” part. The completion A algebra by Lemmas 3.2.7 and 3.2.4. This, by [127, Corollary (4.7.16)], means that  admits a faithful ∗-representation, say π, on a Hilbert space H. From [52, p. 100, A Theorem (26.13)(b)], we then have that  π(x) ≤ pA (x) ≤ x, ∀ x ∈ A.  will be denoted The induced by π and the operator norm of B(H), C ∗ -norm on A, by  · π ; that is,  xπ := π(x), ∀ x ∈ A.  Denote by Ch the maximal commutative ∗-subalgebra of A  Let h ∈ H (A). ∗ ∗ containing h. By Lemma 3.2.5, Ch is a C -algebra, under the C -norm r ≡ rCh (spectral radius of Ch equivalent to the norm  · Ch , the restriction of the norm of  to Ch ). Restricting the continuous, faithful ∗-representation π of A  to Ch [r], we A obtain an isometric ∗-isomorphism of Ch [r] in B(H). Hence, (see also Lemma 3.2.2)  hπ = r(h) = hCh = h, ∀ h ∈ H (A).

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3 Generalized B*-Algebras: Functional Representation Theory

 Then, x = h + ik, h, k ∈ H (A)  and (see also the Take now an arbitrary x ∈ A. arguments used for the proof of (3.2.6)) x ≤ h + k = hπ + kπ ≤ 2xπ ≤ 2(hπ + kπ ) = 2(h + k) ≤ 4x.  In this way, A becomes This shows that the ∗-norms ·, ·π are equivalent on A. a pre C ∗ -algebra with respect to the C ∗ -norm  · π (equivalent to  · ) and so the proof is complete.   Theorem 3.2.10 Let A[ · ] be a Banach ∗-algebra with identity e. Then, A is a C ∗ -algebra under a C ∗ -norm equivalent to the given norm  · , if and only if, the following conditions hold: (i) the collection B∗A has a greatest member; (ii) A is symmetric. Proof “Only if ” part. The condition (i) is proved, in a similar way, as (i) in Theorem 3.2.9. The condition (ii) follows from the fact that every C ∗ -algebra is symmetric (cf. [127, Theorem (4.8.9)]). “if ” part. Define |x| := max{x, x ∗ }, x ∈ A. Then, |x ∗ | = |x|, x ≤ |x|, x ∗  ≤ |x|, ∀ x ∈ A. It follows that if B is a subset of A, such that B ∗ = B, then B is  · -bounded, if and only if, it is | · |-bounded. Therefore, B∗A[·] = B∗A[|·|] . Thus, A[| · |] satisfies the conditions (i) and (ii) of Theorem 3.2.9. We remark that condition (ii) of Theorem 3.2.9 follows from the symmetry of A, which is a completely algebraic property (see beginning of Sect. 3.1 and proof of the “only if ” part of Theorem 3.2.9). This implies that A is a pre C ∗ -algebra, under a C ∗ -norm equivalent to | · |. In [127, p. 181], a Banach ∗-algebra A[ · ] endowed with a C ∗ -norm is called an A∗ -algebra. In this case, the involution on A is continuous with respect to both norms (ibid., Theorem (4.1.15)). Then, it turns out that the norms  · , | · | are equivalent, therefore A[| · |] is a C ∗ -algebra and this completes the proof of the “if ” part.   Remark 3.2.11 Note that the conditions (i), (ii) in Theorems 3.2.9 and 3.2.10 are necessary. Indeed, consider the Banach ∗-algebra A ≡ C (n) [0, 1] of all ncontinuously differentiable functions on the unit interval [0, 1]. Then, A fulfills the condition (ii) of Theorems 3.2.9 and 3.2.10, but not the condition (i). So, if the theorems are true without assuming condition (i), A would be a C ∗ -algebra, which is not true. On the other hand, the disc algebra A(D), D = {z ∈ C : |z| ≤ 1} is a Banach ∗-algebra that fulfills condition (i), but not condition (ii). So, if Theorems 3.2.9 and 3.2.10 are true by removing condition (ii), then A(D) would be a C ∗ -algebra, which is a contradiction.

3.3 GB*-Algebras

51

Recall that Sect. 3.1, together with the ideas of Theorem 3.2.10, lead to the definition of GB∗ -algebras (subject matter of this book), in the next section.

3.3 GB*-Algebras The algebras we introduce in this section generalize the celebrated C ∗ -algebras. They were initiated by G.R. Allan, in 1967 (see [5]) and they were studied first by himself and later by his student P.G. Dixon in [48, 49]. Their significance is due to the fact that they are algebras of unbounded operators as P.G. Dixon showed in [48], therefore they play an important role, in mathematical physics. The following definition stated for a locally convex ∗-algebra A[τ ] with identity, uses Definition 2.2.2 stated for arbitrary locally convex algebras, so naturally an extra condition related to the involution of A and a second one related to the identity element is added (see also comments before Lemma 3.2.1). Definition 3.3.1 Given a locally convex ∗-algebra A[τ ] with identity e, denote by B∗A the collection of all subsets B in A[τ ] that satisfy the following properties: (i) B is absolutely convex, closed and bounded; (ii) e ∈ B, B 2 ⊆ B and B ∗ = B, where B ∗ := {x ∗ : x ∈ B}.  When A is endowed with two different locally convex ∗-algebra topologies τ, τ , then for distinction, we shall use the symbols B∗A[τ ] , B∗A[τ ] respectively, for the preceding collections of subsets of A. For B ∈ B∗A , recall the algebra A[B] = {λx : λ ∈ C, x ∈ B}, generated by B (cf. (2.2.1)). We note that in this case, A[B] is a normed algebra with continuous involution, as it easily follows from the fact that B = B ∗ . Hence, each A[B], B ∈ B∗A , is a normed ∗-algebra. By the standard locally convex space theory, it is evident that the topology τ of A restricted to A[B], B ∈ B∗A , is coarser than the ∗-normed topology induced on A[B] by the gauge function  · B , B ∈ B∗A (see (2.2.2)); that is, τ A[B] ≺  · B , ∀ B ∈ B∗A .

(3.3.7)

According to the preceding, G.R. Allan defined in [5, (2.5) Definition] a GB∗ algebra (abbreviation of generalized B ∗ -algebra) as follows.

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Definition 3.3.2 (Allan) A GB∗ -algebra is a locally convex ∗-algebra A[τ ] with an identity element, such that: (i) B∗A has a greatest (⇔ maximal) member (under the partial ordering of B∗A by inclusion), denoted by B0 ; (ii) A[τ ] is symmetric (in the sense of Definition 3.1.6); (iii) A[τ ] is pseudo-complete. Occasionally, we shall also use the term GB∗ -algebra over B0 , instead of just GB∗ -algebra. P.G. Dixon introduced in [48, (2.5) Definition] a more general than Allan’s Definition 3.3.2, by which one also can have examples of GB∗ -algebras that are not locally convex. For this purpose, he used a slightly different collection than B∗A of Allan. More precisely, if A[τ ] is a topological ∗-algebra with identity e, let BA (or BA[τ ] , for taking into account the given topology), denote the collection of subsets B of A with the properties: (i) B is closed and bounded; (ii) e ∈ B, B 2 ⊂ B, B ∗ = B. According to Dixon [48, p. 694], a topological ∗-algebra is symmetric if fo every x ∈ A the element e + x ∗ x has an inverse in A0 (see also Definition 3.1.6). Then, the author sets the following Definition 3.3.3 (Dixon) A GB∗ -algebra is a topological ∗-algebra A[τ ], such that: (i) BA has a greatest member B0 that is absolutely convex; (ii) A[τ ] is symmetric; (iii) A[B0 ] is complete. Remark 3.3.4 (1) For every locally convex ∗-algebra A[τ ] with identity, the condition (i) of Definition 3.3.2 is equivalent to the condition (i) of Definition 3.3.3. Since this holds, then B∗A and BA have the same greatest member. τ Indeed, let B ∈ BA . Denote by (B) the closed, absolutely convex hull of B. Then, by the properties of B, the separate continuity of multiplication τ of A and the continuity of the involution, we conclude that (B) ∈ BA , but τ ∗ ∗ also (B) ∈ BA . Thus, if BA has a greatest member B0 , we shall have that τ B ⊂ (B) ⊂ B0 . This happens for every B ∈ BA , so that B0 is the greatest member in BA . If now BA has a greatest member B0 , which is absolutely τ convex, then B0 = (B0 ) ∈ B∗A , and so B0 is the greatest member in B∗A . (2) In the case of locally convex ∗-algebras, the notion of a GB∗ -algebra of Dixon (Definition 3.3.5) is weaker than that of a GB∗ -algebra of Allan, because the condition (iii) of Definition 3.3.3 does not imply the condition (iii) of Definition 3.3.2. Indeed, let A[τ ] be a locally convex ∗-algebra and let B0 be

3.3 GB*-Algebras

53

the greatest member in B∗A . Then, by (1) B0 is also the greatest member in BA . Suppose that A[B0 ] is complete. Then, A[B] is complete, for every B ∈ B∗A , but it is not necessarily complete, for every B ∈ (B0 )A (see Definition 2.2.2); that is, A[τ ] is not pseudo-complete, when we refer to the collection (B0 )A (cf. discussion before Proposition 2.2.5). We show now that if B0 is the greatest member in B∗A and A[B0 ] is complete, then A[B] is complete, for every B ∈ B∗A . Indeed, take an arbitrary element B in B∗A . For a Cauchy sequence (xn ) in A[B], we have that, for each ε > 0, there exists n0 ∈ N, such that xm − xn B
0 and some ν ∈ . This yields that D(p ) is τ -dense in A, therefore A[τ ] is a C ∗ -like locally convex ∗-algebra. (iii) ⇒ (ii) This is immediate from the fact that e ∈ B0 and A[B0 ] is a C ∗ -algebra containing the elements (e + h2 )−1 , h∗ = h ∈ A (see Theorem 3.3.9). Thus, the proof is complete.   The following corollary is an immediate consequence of Theorem 3.5.3. Corollary 3.5.4 For every C ∗ -like locally convex ∗-algebra A[τ ], one has that A[B0 ] = D(p ).

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75

Remark 3.5.5 (1) Theorem 3.5.3 implies that a C ∗ -like locally convex ∗-algebra is independent of the method of taking the C ∗ -like families. (2) If A[τ ] is a locally convex ∗-algebra with a C ∗ -like family = {pν }ν∈ of  ] of A[τ ] seminorms, such that D(p ) is τ -dense in A, then the completion A[τ is a C ∗ -like locally convex ∗-algebra. Hence, the assumption of completeness of the locally convex ∗-algebra in Theorem 3.5.3 is not essential. (3) Another proof of (i) ⇒ (iii) in Theorem 3.5.3 has been given by S. J. Bhatt in [17, Theorem 2] using the extended Vidav–Palmer theorem of Wood [156] (see Theorem 7.5.49, in Sect. 7.5 of this book). (4) Let A[τ ] be a locally convex ∗-algebra and (A0 )h := {x ∈ A0 : x ∗ = x}. Denote by B(A) the ∗-subalgebra of A generated by (A0 )h . If A[τ ] is a GB∗ -algebra, the C ∗ -algebra A[B0 ] contains the set (A0 )h (see Corollary 3.3.8), so that we obtain B(A) ⊆ A[B0 ]. For the inverse inclusion use the fact that every element x in A[B0 ] is bounded and that x = h + ik, with h, k self-adjoint elements in A[B0 ]. Hence finally, A[B0 ] = B(A). Suppose now that A[τ ] is a C ∗ -like locally convex ∗-algebra; then by Corollary 3.5.4, D(p ) = A[B0 ]. We can now state the following Proposition 3.5.6 For every C ∗ -like locally convex ∗-algebra, one has the equality B(A) = A[B0 ] = D(p ).  We now compare the C ∗ -algebra A[B0 ] with the Banach ∗-algebra D(p ) in the case of a GB∗ -algebra A[τ ]. Let A[τ ] be a complete GB∗ -algebra with jointly continuous multiplication, where τ is given by a directed family = {pν }ν∈ of ∗-seminorms. Following the same strategy as in the second part of the proof in [60, Theorem 10.23, p. 133], we show that D(p ) is a Banach ∗-algebra, under the norm p . In fact, recalling that A[τ ] has an identity and following the comments before and after (3.5.27), we conclude that D(p ) = A[B], with B ≡ U (p ) ∈ B∗A . Since B ⊂ B0 , one obtains that the Banach ∗-algebra D(p ), under the ∗-norm p , is contained in the C ∗ -algebra A[B0 ], under the C ∗ -norm  · B0 . Summing up, we attain D(p ) ⊆ A[B0 ] and xB0 = p (x), ∀ x ∈ D(p ).

(3.5.29)

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3 Generalized B*-Algebras: Functional Representation Theory

We consider now when D(p ) = A[B0 ]. It is easily seen that the following statements are equivalent: (i) (ii) (iii) (iv)

A[B0 ] = D(p );  · B0 = p ; p is a C ∗ -norm on D(p ); there exists a constant δ > 0, such that pν (x) ≤ δ, for all ν ∈  and x ∈ B0 .

Indeed, the equivalence of (i), (ii), (iii) follows easily from C ∗ -algebra theory (see, e.g., [144, p. 22, Proposition 5.3]). The implication (ii) ⇒ (iv) is trivial. Now (iv) implies that A[B0 ] ⊆ D(p ), while (3.5.29) gives the inverse inclusion. Thus, (iv) ⇒ (i). Remark 3.5.7 (1) The set B0 in B∗A is τ -bounded, therefore for each ν ∈ , there exists a constant δν > 0, depending on ν, such that pν (x) ≤ δν , for all x ∈ B0 . The statement (iv) above, asserts a constant δ > 0, but this does not depend on ν. (2) Note that D(p ) depends on the family of ∗-seminorms = {pν }ν∈ defining the topology τ of A. Indeed, suppose that A[τ ] is a GB∗ -algebra, where τ is defined by an increasing sequence {pn }n∈N of ∗-seminorms. Put pn := npn , n ∈ N. Then, the family {pn }n∈N defines equivalently the given topology τ on A. But,

) is the Banach ∗-algebra corresponding to (A, {p } if D(pN n n∈N ), then clearly,

) ⊂ D(pN ). D(pN =

(3) Note that when A[τ ] is a pro-C ∗-algebra, we have B0 = {x ∈ A : p (x) ≤ 1} ≡ U (p ) (see (1) in Examples 3.3.16), for any family of C ∗ -seminorms defining the topology τ on A. Hence, in this case, we denote D(p ) and p by Ab and  · b , resp. The C ∗ -algebra Ab [ · b ] is called bounded part of A[τ ].  The same happens when A[τ ] is a C ∗ -like locally convex ∗-algebra, as follows from Corollary 3.5.4. Thus, from now on, we shall also use the notation Ab ,  · b , respectively, instead of D(p ), p , for a C ∗ -like locally convex ∗-algebra A[τ ]. (4) In case of a general GB∗ -algebra A[τ ] with jointly continuous multiplication, there does not necessarily exist a family = {pν }ν∈ of ∗-seminorms defining

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77

the topology τ , such that D(p ) = A[B0 ]. Even if there exists a family with subindex, say  , the equality D(p ) = A[B0 ] does not necessarily hold for 

(see (2) above). Thus, a family = {pν }ν∈ of ∗-seminorms defining the topology τ , such that D(p ) = A[B0 ] will be called natural and the corresponding algebra A[τ ] will be called a GB∗ -algebra with a natural family of ∗-seminorms. (5) Suppose that the multiplication on a GB∗ -algebra A[τ ] is separately continuous. Let {pν }ν∈ be an upwards directed family of ∗-seminorms defining the GB∗ topology τ on A. Then, A[B0 ] is a C ∗ -algebra, but D(p ) is not even an algebra, in general. Combining Proposition 3.5.6 with Remark 3.5.7(3), we have the following Proposition 3.5.8 In every C ∗ -like locally convex ∗-algebra A[τ ], we have that B(A) = A[B0 ] = Ab . We shall finish this section with a result on commutative C ∗ -like locally convex ∗-algebras and some examples. A ∗-algebra of functions (see Definition 3.4.8) consisting of C∗ -valued continuous functions on a compact space, is said to be C ∗ -like if it is a C ∗ -like locally convex ∗-algebra. By Theorem 3.5.3 and Theorem 3.4.9 we have the following Theorem 3.5.9 Every commutative C ∗ -like locally convex ∗-algebra is isomorphic to a C ∗ -like algebra of C∗ -valued continuous functions on a compact space. For other results concerning C ∗ -like locally convex ∗-algebras, see end of Sect. 5.3 and Chap. 8. We now present some examples of C ∗ -like locally convex ∗-algebras. Examples 3.5.10 (1) Every pro−C ∗ -algebra A[τ ] is a C ∗ -like locally convex ∗-algebra with B0 = U ( · b ). (2) In particular, the function algebra C(R) of all C-valued continuous functions on R equipped with the family = {pn }n∈N of C ∗ -seminorms, such that pn (f ) =

sup |f (t)|, f ∈ C(R)

t ∈[−n,n]

is a pro-C ∗ -algebra with C(R)b [·b ] = Cb (R), the C ∗ -algebra of all continuous bounded functions on R and f b = supt ∈R |f (t)|, f ∈ Cb (R). (3) Let (hn ) be a sequence of continuous C∗ -valued functions on a compact space X, such that 1 ≤ h1 ≤ h2 ≤ · · · and for all n ∈ N, there exists n1 ∈ N in such a way that hn+1 ≤ h2n ≤ hn1 , where 1 is the constant function 1. We denote by Fhn the set of all continuous C∗ -valued functions f on X, such that sup{|f (x)|/ hn (x) : x ∈ X} < ∞. Since Fhn ⊂ Fhn+1 and h2n ≤ hn1 , for all

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3 Generalized B*-Algebras: Functional Representation Theory

n ∈ N, it follows that F ({hn }) ≡

∞ 

Fhn is a ∗-algebra containing C(X). Since

n=1

hn+1 ≤ h2n , for all n ∈ N, we can define a sequence = {pn } n∈N of seminorms  h1 (x)  on F ({hn }) by pn (f ) = sup e− n |f (x)| : x ∈ X , n ∈ N. Then, we have pn (fg) ≤ p2n (f )p2n (g), pn (f )2 ≤ pn (f ∗ f ), pn (1) ≤ 1, for all n ∈ N and f, g ∈ F ({hn }). Further, F ({hn })b [ · b ] = C(X) and f b = sup |f (x)|, f ∈ F ({hn })b [ · b ]. x∈X

({hn }) of F ({hn }) is a C ∗ -like locally convex ∗Hence, the completion F algebra. (4) Let (X, B, μ) be a finite measure space. Then, the function algebra Lω (X, B, μ) ≡



Lp (X, B, μ)

1≤p 0 : if xB0 < ε, then |f x(e + h2 ) | ≤ 1.

(4.2.4)

If 0 ≤ k ≤ h, then k(e + h2 )−1 ∈ A0 (see Lemma 3.3.7(ii), Theorem 3.3.9(i) recalling that the multiplication of A[τ ] is separately continuous) and k(e + h2 )−1 B0 ≤ h(e + h2 )−1 B0 ≤ 1,

86

4 Commutative Generalized B*-Algebras: Functional Calculus and Equivalent. . .

where the last inequality follows from [127, Lemma (4.8.13)]. Therefore, since  2ε k(e + h2 )−1 B0 < ε, from (4.2.4) we conclude that ! ε ! ! ε !! ! ! ! !f k ! = !f k(e + h2 )−1 (e + h2 ) ! ≤ 1. 2 2 This is true for every k ∈ A+ , with 0 ≤ k ≤ h. Consequently, f (k) ≤ 2ε , for all 0 ≤ k ≤ h, and thus g(h) ≤ 2ε , implying that g(h) < +∞. It is clear that g(αk) = αg(k), for all α ≥ 0 and k ∈ A+ . We now show that g is additive. Let h = h1 + h2 , where h1 and h2 are in A+ . It is easy to verify that g(h1 ) + g(h2 ) ≤ g(h). It remains to prove that g(h) ≤ g(h1 ) + g(h2 ). Suppose first that h is invertible, and 0 ≤ k ≤ h. Then, since A is commutative, 0 ≤ h−1 h1 k ≤ h1 and 0 ≤ h−1 h2 k ≤ h2 (this follows easily from the  and so identification of A with A),

 f (k) = f (h−1 hk) = f h−1 (h1 + h2 )k = f (h−1 h1 k) + f (h−1 h2 k) ≤ g(h1 ) + g(h2 ), ∀ 0 ≤ k ≤ h. Therefore, g(h) ≤ g(h1 ) + g(h2 ), so that g(h) = g(h1 ) + g(h2 ) if h is invertible. Suppose now that h = h1 + h2 , with h1 and h2 in A+ , is not invertible. Using again the functional representation of A[τ ] (Theorem 3.4.9) we conclude that the positive elements h + 2e, h1 + e and h2 + e of A (Definition 4.1.3) are invertible in A and their inverses belong to A0 . Thus, from the previous paragraph we have that g(h) + g(2e) = g(h + 2e) = g(h1 + e) + g(h2 + e) = g(h1 ) + g(h2 ) + 2g(e) = g(h1 ) + g(h2 ) + g(2e). Hence, g(h) = g(h1 ) + g(h2 ) and this holds for all h ∈ A+ . Any h ∈ H (A) can be expressed as a difference of positive elements, i.e. h = h+ − h− , where h+ and h− are in A+ (Theorem 4.1.4). Define g1 : H (A) → R : g1 (h) = g(h+ ) − g(h− ), ∀ h ∈ H (A). We show that g1 is well defined. Suppose that − − + + − h = h+ − h− = h+ 1 − h1 . Then, h + h1 = h1 + h .

4.2 Positive Linear Functionals

87

Since g is additive, − + + + − − g(h+ + h− 1 ) = g(h ) + g(h1 ) and g(h1 + h ) = g(h1 ) + g(h ).

Furthermore, since + − g(h+ + h− 1 ) = g(h1 + h ),

it follows that + − g(h+ ) + g(h− 1 ) = g(h1 ) + g(h ).

Therefore, − g(h+ ) − g(h− ) = g(h+ 1 ) − g(h1 ),

which proves that g1 is well defined. The map g1 is clearly a real valued linear functional, which extends g. If x ∈ A, then x = h + ik, where h, k ∈ H (A). We define now a function f + : A → C : f + (x) = g1 (h) + ig1 (k). Clearly, f + is a linear functional on A extending g1 . Moreover, f + and f − := f + − f are positive linear functionals by using (4.2.3) and properties of g(h). This completes the proof.   For a noncommutative version of Theorem 4.2.5, see Corollary 6.3.16. Corollary 4.2.6 Let A[τ ] be a commutative GB∗ -algebra. Then, there are sufficient positive linear functionals on A to separate its points. Proof A[τ ] is a locally convex space, therefore by Hahn–Banach theorem, there are enough continuous linear functionals on A[τ ] to separate its points. On the other hand, every continuous linear functional on A[τ ] is a linear combination of self-adjoint continuous linear functionals on A[τ ]. Hence, the result follows by the preceding Theorem 4.2.5.   Corollary 4.2.7 Let A[τ ] be a commutative GB∗ -algebra. Then, every continuous linear functional f on A, is a linear combination of (not necessarily continuous) positive linear functionals. Proof Every continuous linear functional is a linear combination of self-adjoint continuous linear functionals. Hence, the result follows from Theorem 4.2.5.   Remark 4.2.8 According to the comments before Theorem 4.2.5, there is no analogue of Corollary 4.2.6 in the noncommutative case. Nevertheless, we may say that a partial answer to this problem is given by Theorem 6.3.4 (Dixon), that plays

88

4 Commutative Generalized B*-Algebras: Functional Calculus and Equivalent. . .

an essential role in the proof of the noncommutative Gelfand–Naimark type theorem for GB∗ -algebras (see Theorem 6.3.5). In this regard, see also [155, Proposition 3.16], where Theorem 4.2.5 is proven for a particular case of a non necessarily commutative Fréchet GB∗ -algebra. Lemma 4.2.9 Let A[τ ] be a commutative GB∗ -algebra and h ∈ A+ . Then, there is an increasing sequence (hn )n∈N of positive elements in A+ ∩ A0 , such that for every positive linear functional f on A, one has f (hn ) → f (h), n ∈ N. Proof Let h ∈ A+ . The following functions are then defined in C(M0 ): ψn (ϕ) :=

  h(ϕ), n,

when  h(ϕ) ≤ n , n ∈ N, ϕ ∈ M0 . when  h(ϕ) > n

It is then clear that ψn (ϕ) ≤ ψn+1 (ϕ) ≤  h(ϕ), ∀ ϕ ∈ M0 , n ∈ N. In particular, each one of ψn is continuous, such that ψn (ϕ) ∈ [0, n], ϕ ∈ M0 , n ∈ N. Now, from Definition 4.1.3 and relation (3.4.16), it follows that ψn is the Gelfand transform of a unique element hn in A+ ∩ A0 , n ∈ N. From the preceding, it is evident that 0 ≤ h1 ≤ h2 ≤ · · · ≤ hn ≤ · · · ≤ h. Thus, considering a positive linear functional f on A, we obtain that 0 ≤ f (h1 ) ≤ f (h2 ) ≤ · · · ≤ f (hn ) ≤ · · · ≤ f (h). Hence, supn f (hn ) < +∞, so that setting g(h) := supn f (hn ), we have 0 ≤ g(h) ≤ f (h).

(4.2.5)

We shall show that   g(h) = sup f (u) : u ∈ A+ ∩ A0 , 0 ≤ u ≤ h < ∞.

(4.2.6)

Indeed, let α ≡ sup{f (u) : u ∈ A+ ∩ A0 , 0 ≤ u ≤ h}. Then, α < +∞ and α ≥ g(h). On the other hand, since u ∈ A+ ∩ A0 with 0 ≤ u ≤ h, there will be m ∈ N, such that u ≤ hn , with n ≥ m.

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89

This implies that α ≤ g(h), so that (4.2.6) is proved. It is then clear that g(h) = f (h), when h ∈ A+ ∩ A0 .

(4.2.7)

Now arguing as in the proof of Theorem 4.2.5, it can be shown that g is additive and positively homogeneous on A+ , so that it is extended to a real linear functional on the real vector space H (A). Because of (3.1.1), we further obtain an extension of the latter to a positive (see (4.2.5)) linear functional on A, which we also denote by g. Thus, defining f0 (x) := f (x) − g(x), x ∈ A, we obtain a positive linear functional f0 on A, as easily follows from (3.1.1) and (4.2.5), while f0 vanishes on A+ ∩ A0 by (4.2.7). It is also evident that f0 (e) = 0, so by the Cauchy–Schwarz inequality (Proposition 4.2.2(ii) ), we obtain that f0 (x) = 0, for all x ∈ A, therefore f (x) = g(x), ∀ x ∈ A. It it clear now that f (h) = g(h) = sup f (hn ) = lim f (hn ), h ∈ A+ ; n

this completes the proof.

n

 

Corollary 4.2.10 Let A[τ ] be a commutative GB∗ -algebra and x ∈ A. Then, there is a sequence (xn )n∈N in A0 , such that f (xn ) → f (x), for every positive linear functional f on A. Proof Let x ∈ A. Then, x = u+iv with u, v ∈ H (A). Applying Theorem 4.1.4(ii), we have that u = u+ − u− and v = v + − v − , with u+ , u− and v + , v − elements from A+ . Then, Lemma 4.2.9 provides us with increasing sequences (uιn )n∈N and (vnι )n∈N , ι = +, − from A+ ∩ A0 , such that f (uιn ) → f (uι ) and f (vnι ) → f (v ι ), ι = +, −, for any positive linear functional f on A. The assertion now − + − follows by setting xn := u+   n − un + i(vn − vn ), n ∈ N. Theorem 4.2.11 (Bhatt) Let A[τ ] be a GB∗ -algebra. Then, A[B0 ] is sequentially dense in A[τ ]. Proof Let h ∈ H (A) and C be the closed ∗-subalgebra of A[τ ] generated by h and the identity element e ∈ A. Then, C is a commutative GB∗ -algebra (Proposition 3.3.19). Since h is an arbitrary self-adjoint element of A, we may suppose, without loss of generality, that A[τ ] is commutative. Now, Theorem 3.3.9 and Lemma 3.4.2(i) yield that A[B0 ] is a C ∗ -algebra, such that all three elements e, (e+ h2 )−1 and h(e + h2 )−1 belong to it. Further, through the Gelfand representation,

90

4 Commutative Generalized B*-Algebras: Functional Calculus and Equivalent. . .

A[B0 ] is isometrically ∗-isomorphic to the function C ∗ -algebra C(M0 ), M0 being the Gelfand space of A[B0 ]. Thus, we conclude that h(e + h2 )−1 B0 ≤ 1, which implies that h(e + h2 )−1 ∈ B0 . Since, for every positive real number β, β 1/2h ∈ H (A), we therefore have that −1

β 1/2 h(e + βh2 )−1 = β 1/2 h e + (β 1/2h)2 ∈ B0 .

(4.2.8)



h − h(e + βh2 )−1 = h e − (e + βh2 )−1 = βh3 (e + βh2 )−1 .

(4.2.9)

Moreover,

Now, the multiplication of A[τ ] is separately continuous; hence, for each 0neighbourhood V in A, there is a 0-neighbourhood U in A, such that h2 U ⊂ V . Furthermore, since B0 is a bounded set, there is a positive number α, such that α 1/2 B0 ⊂ U . Taking also into account (4.2.8) and (4.2.9) with β = α, we conclude that for sufficiently small α,

−1 h − h(e + αh2 )−1 = α 1/2 h e + (α 1/2 h)2 α 1/2 h2 ∈ α 1/2 h2 B0 ∈ h2 U ⊂ V , therefore h = lim h(e + αh2 )−1 . α→0+

(4.2.10)

Observe now that h(e + αh2 )−1 =

1 α 1/2

−1

(α 1/2 h) e + (α 1/2 h)2 ∈ H (A[B0 ]), α > 0,

which, by virtue of (4.2.10) and (3.1.1), shows that A[B0 ] is dense in A[τ ], taking α = 1/n, n ∈ N. This completes the proof.   From Theorem 4.2.11 and the fact that A[B0 ] ⊂ A0 , we have the following Corollary 4.2.12 (Allan) Let A[τ ] be a GB∗ -algebra. Then, A0 is dense in A[τ ]. For the proof of Allan for the previous Corollary 4.2.12, see [5, (4.7) Theorem]. Furthermore, from Theorem 4.2.11, Examples 3.3.16(1) and Remark 3.5.7(3) (as an alternative to the latter, see Proposition 3.5.8), we obtain the following (Apostol [8]; see also [60, Theorem 10.23]) Corollary 4.2.13 Let A[τ ] be a pro-C ∗ -algebra. Then, the C ∗ -algebra Ab [ · b ] (bounded part of A[τ ]) is dense in A[τ ]. Notes Proposition 4.2.3(ii) has been proved by P.G. Dixon in [51] (see also [60, p. 191, Theorem 15.5]). Statement (iii) of the same proposition is an immediate consequence of (ii), since every σ -C ∗ -algebra is a Fréchet pro-C ∗-algebra and as a pro-C ∗-algebra has a bounded approximate identity (Inoue; [75, Theorem 2.6], see also [60, p. 137, Theorem 11.5]). Theorem 4.2.11 is due to S.J. Bhatt (cf. [17, (2) Theorem]).

4.3 Equivalent Topologies

91

4.3 Equivalent Topologies The Gelfand–Naimark type theorem for a commutative GB∗ -algebra (see Theorem 3.4.9) is clearly a purely algebraic result, in contrast to the classical Gelfand– Naimark theorem for a commutative C ∗ -algebra (with identity), where the corresponding Gelfand map is an isometric ∗-isomorphism. This is rather due to the fact that the C ∗ -property is directly related to the given normed topology of a given C ∗ -algebra, while the GB∗ -condition on a locally convex ∗-algebra A[τ ] is related to a whole collection (namely, the B∗ -collection) of certain bounded subsets of A[τ ], where the greatest member B0 of this collection is mostly used. In this way, one obtains the C ∗ -algebra A[B0 ], through which the study of the GB∗ structure is realized, as it also happens with the Gelfand–Naimark type theorem for a commutative GB∗ -algebra. Concerning C ∗ -algebras, we know that they have a unique C ∗ -norm. So a natural problem is the following: Question Do we have a similar situation with a GB∗ -topology on a GB∗ -algebra? The answer will be investigated in this section. We shall see that if a commutative ∗-algebra with identity is a GB∗ -algebra under two distinct topologies τ, τ , then these two topologies have to be connected under a certain equivalence relation (see, for example, Corollary 4.3.10). Definition 4.3.1 (Allan) Let A be an algebra, which is a locally convex algebra under two topologies, say τ, τ . Consider the corresponding collections (B0 )1 , (B0 )2 for A[τ ] and A[τ ] respectively, as in Definition 2.2.2. We shall say that τ and τ

are equivalent if the collections (B0 )1 , (B0 )2 fulfil the following properties: (i) for every B1 ∈ (B0 )1 there is some B2 ∈ (B0 )2 , such that B1 ⊆ B2 ; (ii) the same is true by interchanging (B0 )1 , (B0 )2 . In this regard, we have Remark 4.3.2 and Proposition 4.3.3. Remark 4.3.2 (1) The relation introduced in the preceding definition is clearly an “equivalence relation”, that we shall denote by “∼”. For two equivalent topologies τ and τ

on A as before, we shall write τ ∼ τ . (2) Let us start in Definition 4.3.1 with a commutative ∗-algebra A equipped with two topologies τ, τ , under which it becomes a pseudo-complete locally convex ∗-algebra. Let B0 (τ ), B0 (τ ) be the greatest members in B∗A[τ ] and B∗A[τ ] , respectively. Then, it is easily seen that τ ∼ τ , if and only if, B0 (τ ) = B0 (τ ). Proposition 4.3.3 Let A be a commutative ∗-algebra with an identity. Suppose that A is also a locally convex algebra under two topologies τ, τ . Then, the following hold:

(i) suppose that A[τ ] is a GB∗ -algebra and B0 the greatest member in B∗τ ≡  B∗A[τ ] . Then τ ∼ τ , if and only if, B0 is also the greatest member in B∗τ ≡  B∗A[τ ] ;

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(ii) if the involution ∗ is τ -continuous, A[τ ] a GB∗ -algebra and τ ∼ τ , then A[τ ] is also a GB∗ -algebra and both topologies determine the same C ∗ -algebra, which is the commutative C ∗ -algebra A0 . Proof We only sketch the proof of pseudo-completeness of A[τ ] in (ii); everything else follows easily from the very definitions (see also Lemma 3.3.7(ii)). So, let B

be an arbitrary element in B∗τ . We shall prove that the normed ∗-algebra A[B ] is complete. Since τ ∼ τ , we have that for the chosen B ∈ B∗τ there is B ∈ B∗τ , such that B ⊆ B, therefore A[B ] ⊆ A[B] and  · B ≤  · B . Hence, if (xn )n∈N is a Cauchy sequence in A[B ], it will also be so in A[B], which is a Banach algebra. So there is x ∈ A[B], such that xn − xB → 0. Following now similar arguments to those in the proof of Proposition 2.2.6, we obtain the completeness of A[B ].   We shall finish this section by describing those topologies on A that are equivalent to the given topology τ (see Corollary 4.3.10). More precisely, it will be proved that any two topologies τ, τ , that make a commutative ∗-algebra with identity a GB∗ -algebra, have to be equivalent in the sense of Definition 4.3.1.  If A is an algebra, denote by A∗ the algebraic dual of A. Let P(A) be the set of all positive linear functionals of A and Apf be the linear subspace of A∗ generated by P(A). If A[τ ] is a locally convex algebra, let A be the (topological) dual space of A[τ ]; i.e., A := {f ∈ A∗ : f is continuous}. The following is an immediate consequence of Corollary 4.2.7. Remark 4.3.4 Let A[τ ] be a commutative GB∗ -algebra. Then, A ⊂ Apf . Now given an algebra A, take the pair (A, Apf ) with the bilinear form (A, Apf ) # (x, f ) → f (x) ∈ C. Then, according to [74, p. 183, §2, Definition 1], we have a dual system with respect to the preceding bilinear form, in the sense that A separates points of Apf and Apf separates points of A; for the latter, in the case when A[τ ] is a commutative GB∗ algebra, see Corollary 4.2.6. Thus, we may consider the weak topology σ (A, Apf ) on A, which is a (Hausdorff) locally convex space topology defined by the family of seminorms {qf }f ∈Apf , given as follows (see [74, p. 185]) qf (x) := |f (x)|, ∀ x ∈ A.

(4.3.11)

It is then clear that each f ∈ Apf is continuous with respect to the weak topology σ (A, Apf ) on A. In this aspect, we set the following Definition 4.3.5 (Allan) If τ pf ≡ σ (A, Apf ) ≡ σ (A, P(A)), then the weak topology τ pf on A is called positive-functional topology on A.

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From the definition of τ pf , Remark 4.3.4 and the comments just before Definition 4.3.5 we are led to the result that follows. Proposition 4.3.6 Let A[τ ] be a commutative GB∗ -algebra. The following hold: (i) the dual of the locally convex space A[τ pf ] is Apf ; (ii) the weak topology σ (A, A ) on A is coarser than the positive-functional topology τ pf on A; i.e., σ (A, A ) ≺ τ pf . Theorem 4.3.9 below shows that τ ∼ τ pf . The proof is based on a series of Lemmas that are first proved. Lemma 4.3.7 Let A[τ ] and A[τ pf ] be as before. Then, A[τ pf ] is a locally convex ∗-algebra. Proof In the comments after Remark 4.3.4, we noticed that the algebra A[τ pf ] is a locally convex space. It remains to show that it also has a separately continuous multiplication and continuous involution. Let h ∈ A+ and f ∈ P(A). Define the functional fh (x) := f (hx), ∀ x ∈ A. Then, fh is a positive linear functional on A (see Theorem 4.1.4(i) too). Now it easily follows (see (4.3.11)) that the map A[τ pf ] → A[τ pf ] : x → hx is continuous and since any arbitrary y ∈ A is a linear combination of positive elements (see (3.1.1) and Theorem 4.1.4(ii)), we conclude that the map A[τ pf ] → A[τ pf ] : x → yx, y ∈ A, is continuous. This means that the multiplication of A[τ pf ] is separately continuous. The continuity of involution follows from the definition of τ pf and the fact that every f ∈ P(A) is ∗-preserving (Proposition 4.2.2(i) ). Thus, the proof is complete.   Lemma 4.3.8 Let A[τ ], A[τ pf ] be as above and   B1 := x ∈ A : f (x ∗ x) ≤ 1, f ∈ P(A) with f (e) ≤ 1 . Then, B1 has the following properties: (1) it is τ - and τ pf -bounded; (2) it is absolutely convex; (3) B12 ⊆ B1 ; B1∗ = B1 ; and e ∈ B1 . Proof (1) Let f ∈ P(A), such that f (e) ≤ 1. Then, by the Cauchy–Schwarz inequality (see Proposition 4.2.2(ii) ), we have that |f (x)|2 ≤ f (e)f (x ∗ x) ≤ 1, ∀ x ∈ B1 .

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Hence, B1 is τ pf -bounded and since A ⊆ Apf (Remark 4.3.4), it follows that B1 is σ (A, A )-bounded, consequently τ -bounded by [74, p. 209, Theorem 3]. So (1) is shown. (2) Let λ ∈ C with |λ| ≤ 1. Then, clearly λB1 ⊆ B1 , so that the proof will be complete if we still show that B1 is convex. So let λ ∈ [0, 1] and x, y ∈ B1 . Put z := λx + (1 − λ)y and take f ∈ P(A) with f (e) ≤ 1. Applying the Cauchy–Schwarz inequality (Proposition 4.2.2(ii)), we obtain that f (z∗ z) ≤ λ2 f (x ∗ x) + (1 − λ)2 f (y ∗ y) + 2λ(1 − λ)f (x ∗ x)1/2f (y ∗ y)1/2 2

= λf (x ∗ x)1/2 + (1 − λ)f (y ∗ y)1/2 ≤ 1. The last inequality occurs, since x, y ∈ B1 . Hence, z ∈ B1 and B1 is convex. (3) It is easily seen that e ∈ B1 and that B1∗ = B1 . So it remains to be proved that B12 ⊆ B1 . Consider arbitrary elements x, y ∈ B1 and put z := xy. For f ∈ P(A) with f (e) ≤ 1, define the functional fy (w) := f (y ∗ wy), ∀ w ∈ A. It is evident that fy is a positive linear functional on A and either fy = 0, where in this case one has f (z∗ z) = fy (x ∗ x) = 0 < 1, or fy = 0, that is fy (e) = 0, otherwise from the Cauchy–Schwarz inequality we are led to a contradiction. Now, since y ∈ B1 , we clearly have fy (e) = f (y ∗ y) ≤ 1 and since also x ∈ B1 , we conclude that f (z∗ z) = fy (x ∗ x) ≤ 1, which implies z ∈ B1 . Thus, B12 ⊆ B1 .

  A[τ pf ]

Theorem 4.3.9 Let A[τ ], be as above. Then, τ ∼ member B0 in B∗A[τ ] takes the form

τ pf

and the maximal

  B0 = x ∈ A : f (x ∗ x) ≤ 1, f ∈ P(A) with f (e) ≤ 1 . Proof We first show that B0 equals to B1 of Lemma 4.3.5. It is obvious from Lemma 4.3.8 that the τ -closure of B1 belongs to B∗A[τ ] . Hence, B1 ⊆ B0 . Now, note that A0 = A[ · B0 ] is a C ∗ -algebra and its closed unit ball is B0 . Thus, taking an element f ∈ P(A) with f (e) ≤ 1, then (by Proposition 4.2.3(i))

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f restricted to A0 is a continuous positive linear functional, such that f  = f (e) ≤ 1. Moreover, x ∈ B0 implies x ∗ x ∈ B0 , so that f (x ∗ x) ≤ f (e)x ∗ xB0 ≤ 1. Thus, x ∈ B1 and finally B1 = B0 . By Lemma 4.3.8 B0 ∈ B∗A[τ pf ] . We show now that B0 is the greatest member

of B∗A[τ pf ] . So let B be an arbitrary element in B∗A[τ pf ] . Then, B is τ pf -bounded, therefore σ (A, A )-bounded by Proposition 4.3.6(ii), which by [128, p. 67, Theorem 1] yields that B is τ -bounded. It follows that the τ -closure of B belongs to B∗A[τ ] , consequently B ⊆ B0 . Repeating the same argument for the τ pf -closure of B0 in place of B , we conclude that B0 is τ pf -closed. Thus, B0 is the greatest member in B∗A[τ pf ] . Proposition 4.3.3(i) yields now that τ ∼ τ pf ; this completes the proof.  

Corollary 4.3.10 Let A be a commutative ∗-algebra with identity, which is a GB∗ algebra under two topologies τ and τ . Then, τ ∼ τ . Proof Theorem 4.3.9 implies that τ ∼ τ pf and τ ∼ τ pf , therefore τ ∼ τ .

 

For an extension of Corollary 4.3.10 to the noncommutative case, see Corollary 6.3.7 (cf. also the discussion after it). Further, it will be shown that there is a finest locally convex ∗-algebra topology on a given commutative GB∗ -algebra A[τ ] that is equivalent to the initial topology τ . So, taking into account Corollary 4.3.10, we conclude that (see Theorem 4.3.13) there exists a finest topology, under which A is a GB∗ -algebra. To prove the announced result, we need definition of the so-called Mackey topology that we first exhibit. Let E[τ ] be a locally convex space and E its dual. Denote by K the family of all subsets K of E that are absolutely convex and σ (E , E)-compact. Then, we have (see [74, p. 206, Definition 1]) the following Definition 4.3.11 The topology of uniform convergence on the members of the family K is denoted by τ (E, E ) and is called the Mackey topology. Namely, τ (E, E ) is the K-topology on E corresponding to the dual system (E, E ), with respect to the bilinear form x, f := f (x), x ∈ E, f ∈ E . The Mackey topology is a locally convex topology induced by the seminorms qK , K ∈ K, given by (cf. [74, p. 195])   qK (x) := sup |x, f | : f ∈ K , ∀ x ∈ A.

(4.3.12)

One important property of the Mackey topology τ (E, E ) reads as follows: If τ is a locally convex topology on E, the dual of E[τ ] equals to E , if and only if, σ (E, E ) ≺ τ ≺ τ (E, E );

(4.3.13)

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for this and further results on the Mackey topology, see [74, p. 206, Proposition 4 and p. 203, §5]. Lemma 4.3.12 Let A[τ ] be a locally convex algebra with dual A . Let τK := τ (A, A ). Then, A[τK ] is a locally convex algebra and τK ∼ τ . Proof We only show that the multiplication in A[τK ] is separately continuous. Everything else holds since A[τK ] is a topological vector space (cf. [74, p. 195, §4]). Thus, let us take a fixed point y ∈ A, and prove that the map A[τK ] → A[τK ] : x → xy is continuous. Consider the map Ty : A → A : f → Ty (f ) := fy , with fy (x) := f (xy), x ∈ A. Clearly Ty is a well defined σ (A , A) − σ (A , A) continuous linear operator on A . Hence, Ty (K) ∈ K, for every K ∈ K. On the other hand, if K ◦ is the polar of K ∈ K, i.e.,   K ◦ := x ∈ A : |f (x)| ≤ 1, ∀ f ∈ K , then by the very definitions it is easily seen that ∀ x ∈ A, xy ∈ K ◦ ⇔ x ∈ Ty (K)◦ . For a fixed y ∈ A, this proves the continuity of the map A[τK ] → A[τK ] : x → xy, since finite intersections of the sets λK ◦ , λ > 0, K ∈ K, form a fundamental system of neighbourhoods of 0 for the topology τK (cf. [74, p. 195, §4]). Thus, the multiplication in A[τK ] is separately continuous. Taking now an absolutely convex set in A, this is τ -bounded and τ -closed, if and only if, it is τK -bounded and τK -closed (see [35, Chap. IV, §2, Proposition 4, Corollaire 1 and Théorème 3]). Consequently, the corresponding collections BA[τ ] and BA[τK ] are the same, so that τ ∼ τK from Definition 4.3.1. This completes the proof.   For the notation applied to the next theorem, see discussion before Definition 4.3.5. Theorem 4.3.13 Let A[τ ] be a commutative GB∗ -algebra. Then, the topology τ $ := τ (A, Apf ) is the finest locally convex ∗-algebra topology on A that is equivalent to τ . In particular, if τlc∗ is any locally convex ∗-algebra topology on A, such that all the positive linear functionals on A[τlc∗ ] are continuous, then τlc∗ ∼ τ, if and only if, τ pf ≺ τlc∗ ≺ τ $ . Proof By Theorem 4.3.9 τ ∼ τ pf , therefore we may think of A[τ ] as being A[τ pf ], whose dual space is Apf . Hence, from Lemma 4.3.12, we have that A[τ $ ] is a locally convex algebra and τ $ ∼ τ pf . Thus, τ $ ∼ τ . We must also show that A[τ $ ] has a continuous involution.

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The involution ∗ on A gives rise to a well-defined, continuous, conjugate linear map Apf [σ (Apf , A)] → Apf [σ (Apf , A)] : f → f ∗ , with x, f ∗ := x ∗ , f , x ∈ A, f ∈ Apf .

If K is an absolutely convex, σ (Apf , A)-compact subset of Apf , its image K ∗ := {f ∗ : f ∈ K} under the preceding (dual) map is also an absolutely convex, σ (Apf , A)-compact subset of Apf , so that K ∗ ∈ K and   qK (x ∗ ) := sup |x ∗ , f | : f ∈ K = sup{|x, f ∗ | : f ∗ ∈ K ∗ } =: qK ∗ (x), ∀ x ∈ A.

This shows that the involution on A is τ $ -continuous. Let now τlc∗ be an arbitrary locally convex ∗-algebra topology on A equivalent to the given one, i.e., τlc∗ ∼ τ . Then, τlc∗ ∼ τ pf from Theorem 4.3.9, A[τlc∗ ] is a (commutative) GB∗ -algebra from Proposition 4.3.3(ii) and its dual A[τlc∗ ]

is contained in the dual Apf of A[τ pf ], according to Remark 4.3.4. Hence, A[τlc∗ ] [σ (A[τlc∗ ] , A)] is contained in Apf [σ (Apf , A)], therefore a weak∗ -compact subset of the dual A[τlc∗ ] of A[τlc∗ ] is also weak∗ -compact as a subset of the dual Apf [σ (Apf , A)] of A[τ pf ]. In conclusion (see also (4.3.13)), we obtain τlc∗ ≺ τ (A, A[τlc∗ ] ) ≺ τ (A, Apf ) = τ $ . This shows that τ $ is the finest locally convex ∗-algebra topology on A, equivalent to τ . For the last claim of the theorem, let σ be an arbitrary locally convex ∗-algebra topology on A, such that every positive linear functional on A[σ ] is continuous. Then, Definition 4.3.5 implies that τ pf ≺ σ . On the other hand, if σ ∼ τ , then as in the previous paragraph we conclude that σ ≺ τ $ ; consequently, τ pf ≺ σ ≺ τ $ . Conversely, if σ is an arbitrary locally convex ∗-algebra topology on A, such that every positive linear functional on A[σ ] is continuous and τ pf ≺ σ ≺ τ $ , then we obtain that the dual of A[σ ] coincides with Apf , so [35, Chap. IV, §2, Proposition 4 and Théorèm 3] yields that BA[σ ] = BA[τ pf ] (see also the last part of the proof of Lemma 4.3.12). It follows that σ ∼ τ pf and since τ pf ∼ τ by Theorem 4.3.9, one finally obtains σ ∼ τ and this completes the proof.   We close this section by proving that given a commutative GB∗ -algebra A[τ ], the space A[τ $ ] is barrelled and bornological (Theorem 4.3.16). For this we need some extra definitions and results from the theory of locally convex spaces that we first exhibit (see, e.g., [74, Chapter 3, pp. 208 and 210, as well §6 and §7, respectively], but also [35, 123, 131]). Definition 4.3.14 Let E[τ ] be a locally convex space. (1) An absolutely convex subset of E that is absorbing and τ -closed is said to be a barrel. If every barrel is a 0-neighbourhood, then E[τ ] is said to be a barrelled space.

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(2) An absolutely convex subset of E that absorbs every τ -bounded subset of E is called bornivorous. If every bornivorous subset of E is a 0-neibhourhood, then E[τ ] is called a bornological space. Note that every metrizable and complete locally convex space (i.e., a Fréchet space, hence a Banach space too) is barrelled, while a metrizable locally convex space is bornological (cf., for instance, [74, p. 214, Corollary and p. 222, Proposition 3, resp.]. In particular, a locally convex space E[τ ] is bornological, if and only if, τ coincides with the Mackey topology τ (E, E ) (ibid., p. 221, Proposition 2(i)]). Yet, every quasi-complete (i.e., all closed and bounded subsets are complete) bornological locally convex space is barrelled [131, p. 63, Corollary]. Let now E[τ ] be a locally convex space and N the collection of all barrels in E. The collection N can be considered as a base of 0-neighbourhoods for a topology, say τ , on E. Then, E[τ ] is a barrelled locally convex space and τ is called the associated barrel topology, with respect to the given topology τ of E. Then, τ ≺ τ

(apply e.g., [74, p. 87, Proposition 4]) and E[τ ] is barrelled, if and only if, τ = τ . In a similar way, the collection K of all bornivorous subsets of E, gives a base of 0-neighbourhoods for a locally convex topology, say τ

, on E that is called the associated bornological topology, with respect to the topology τ of E. Again, one has that τ ≺ τ

and that E[τ ] is bornological, if and only if, τ = τ

. Using the introduced topologies τ , τ

, we have the following Lemma 4.3.15 Let A[τ ] be a locally convex algebra. Then, A[τ ], A[τ

] are both locally convex algebras. If moreover A[τ ] is a locally convex ∗-algebra, the same is true for both A[τ ] and A[τ

]. Proof First, we must prove that multiplication in A is separately continuous with respect to the topologies τ and τ

; secondly, that if A[τ ] is endowed with a continuous involution, then this is also τ - respectively, τ

-continuous. Let U be a barrel in A[τ ] and y ∈ A. Set U := {x ∈ A : xy ∈ U }. Using the fact that U is a τ -barrel, it is easily seen that U is a τ -barrel too. This proves that the map A[τ ] → A[τ ] : x → xy is continuous, i.e., multiplication in A[τ ] is separately continuous. Suppose now that A[τ ] has a continuous involution ∗. Then, taking a barrel U in A[τ ], the set U ∗ := {x ∗ : x ∈ U } is a τ -barrel and this shows that ∗ : A[τ ] → A[τ ] is continuous. In the same way we work for the case of A[τ

]. Namely, if V is a bornivorous subset in A[τ ] and V := {x ∈ A : xy ∈ V }, y a fixed element in A, then, V

is absolutely convex and if S is a τ -bounded subset in A, the set Sy is also τ bounded, since the map A[τ ] → A[τ ] : x → xy is continuous. It follows that Sy will be absorbed by every 0-neighbourhood, so that there exists α > 0, with αSy ⊆ V , which yields αS ⊆ V . Hence, V absorbs τ -bounded sets, consequently

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99

V is bornivorous. It is now evident that the map A[τ

] → A[τ

] : x → xy is continuous. So A[τ

] is a locally convex algebra. On the other hand, if A[τ ] has a continuous involution and V is a τ -bornivorous set in A, the same is true for the set V ∗ , i.e., involution in A[τ

] is continuous.   Theorem 4.3.16 Let A[τ ] be a commutative GB∗ -algebra and let τ $ := τ (A, Apf ). Then, A[τ $ ] as a locally convex space is barrelled and bornological. Proof Recall the associated barrel and bornological topologies τ , respectively τ

, with respect to the given topology τ on A, introduced just before Lemma 4.3.15. It will be shown that τ = τ $ = τ

. Observe that τ pf ∼ τ (Theorem 4.3.9) and that τ ∼ τ $ (Theorem 4.3.13), consequently τ $ ∼ τ . From Proposition 4.3.3 and Theorem 4.3.13, we conclude that A[τ $ ] is a commutative GB∗ -algebra with the corresponding C ∗ -subalgebra A0 (= A[B0 ]) exactly the same as that of A[τ ] (recall that B0 (τ ) = B0 (τ $ )). Thus, we consider the greatest member B0 in B∗A[τ $ ] and let U be an arbitrary τ $ barrel in A (i.e., U is a 0-neighbourhood in A[τ ]). Then, U is τ $ -closed, therefore U ∩ A0 is  · B0 -closed in the C ∗ -algebra A0 . Since U is absorbing and absolutely convex, the same is true for U ∩ A0 , so that U ∩ A0 is a barrel in A0 . But, A0 as a Banach space is barrelled (cf. comments after Definition 4.3.14), consequently U ∩ A0 is a 0-neighbourhood in A0 and thus absorbs the closed unit ball B0 of A0 . It follows that U absorbs B0 , therefore B0 is τ -bounded. Moreover, B0 is τ $ closed and since τ $ ≺ τ (see discussion before Lemma 4.3.15 about τ ) it is also τ -closed. So B0 ∈ B∗A[τ ] . If now B is an arbitrary member in B∗A[τ ] , then B is τ $ -bounded and its τ $ -closure belongs to B∗A[τ $ ] . Hence, B ⊆ B0 and so B0 is the greatest member in B∗A[τ ] . Thus, from Proposition 4.3.3(i) and Theorem 4.3.13 we obtain τ ∼ τ $ ∼ τ . But (ibid.) τ $ is the finest locally convex ∗-algebra topology on A that is equivalent to the given topology τ and also τ $ ≺ τ as we noticed before. Consequently, by Lemma 4.3.15, we finally obtain τ $ = τ and this proves that A[τ $ ] is barrelled. Suppose again that B0 is the greatest member in B∗A [τ $ ]. Then, using the terms ’bornivorous’ and ’bornological’ instead of ’barrel’ and ’barrelled’ we conclude, as before, that B0 ∈ B∗A[τ

] . For proving that A[τ

] is bornological we argue exactly as in the case of barrelledness above, replacing τ with τ

. So the proof is complete.   For a similar result in the noncommutative case, see Theorem 6.2.6 in Sect. 6.2.

4.4 A *-Algebra of Functions with no GB*-Topology From the information given in Chaps. 3 and 6, we may briefly say that: (1) a GB∗ -algebra A[τ ], whose topology τ is defined by a complete norm, is a C ∗ -algebra (see Corollary 3.3.11(1));

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 of C∗ (2) a commutative GB∗ -algebra A[τ ] is ∗-isomorphic to a ∗-algebra A valued continuous functions on a Hausdorff compact space M0 , that is nothing other than the maximal ideal space of the C ∗ -algebra A[B0 ] = A0 ; moreover,  contains the function C ∗ -algebra C(M0 ) (cf. Theorem 3.4.9); the ∗-algebra A (3) an arbitrary GB∗ -algebra is ∗-isomorphic to a ∗-algebra of unbounded operators in a Hilbert space (see Theorem 6.3.5). In Example 4.4.1 below, it is shown that not every ∗-algebra of functions is realized as in (2) above.  In what follows we use the term “almost everywhere” (abbreviated to a.e.) in the topological sense, meaning that an equality holds almost everywhere, except on a set of first category.

The following example is due to P.G. Dixon [50, p. 160, Example 1], see also [28, XI.10, Theorem 12, Corollary 1], or [XI.7, Exercise 5], in the 3rd Edition of the same book. Example 4.4.1 (Dixon) Let A be the ∗-algebra of all C-valued Borel functions on the unit interval [0, 1], modulo equality a.e., in the preceding topological sense. Denote by 1 the identity element of A. We shall say that a function f ∈ A is essentially bounded if there is a positive number k such that |f (t)| ≤ k, for all t ∈ [0, 1], except on a subset of first category. For f ∈ A being essentially bounded, let   f −1 (k, ∞) ≡ t ∈ [0, 1] : |f (t)| > k and

  Ufess ≡ k a positive number : f −1 (k, ∞) is a set of first category be the set of essential upper bounds of f . Then, as usually, one defines ess sup f := inf Ufess , if Ufess = ∅ and ess sup f := ∞, otherwise. Consider now the ∗-subalgebra of A   Aess b ≡ f ∈ A : f is essentially bounded and put ess f ess b := ess sup f, ∀ f ∈ Ab .

Then, Aess is a C ∗ -algebra endowed with the C ∗ -norm  · ess and it is a ∗b b subalgebra of A, such that for each f in A, the elements (1 + f ∗ f )−1 and f (1 + f ∗ f )−1 belong to Aess b .

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We shall show that A has no GB∗ -topology. Suppose that A admits a GB∗ topology, say τ , with B0 the greatest member in B∗A[τ ] . Then, since A[τ ] is a commutative GB∗ -algebra, it follows that (Lemma 3.3.7(ii)) A[B0 ] = A0 = Aess b , where A0 is the set of all bounded elements of A[τ ]. Using the Gelfand ∗isomorphism (extension of the Gelfand ∗-isomorphism of Aess b to A; cf. Theorem 3.4.9), one can see that the projections (i.e., the self-adjoint idempotents) of A belong to B0 . These projections are exactly the characteristic functions of the Borel subsets of [0, 1]. Take now a non-zero projection p corresponding to a Borel set P ⊆ [0, 1]. Then, one can find a strictly decreasing sequence of nonempty Borel sets ∞P1 , P2 , P3 , . . . in [0, 1], with characteristic functions p1 , p2 , p3 , . . ., such that n=1 Pn = P ; we remind the reader that we work with equivalence classes, with respect to the equivalence relation “a.e. except on a set of first category”. Now, according to the correspondence Pn ↔ p n , n ∈ N and properties of the Pn ’s, the infinite sum in the definition f (t) := ∞ n=1 pn (t), t ∈ [0, 1], is in fact finite. Hence, f belongs to A[τ ]. By the (separate) continuity of multiplication in A[τ ], we have that for each 0-neighbourhood U in A[τ ], there is another 0neighbourhood V in A[τ ], such that f (V ) ⊆ U . On the other hand, since B0 is τ -bounded, there is a positive number N with B0 ⊆ NV , therefore N −1 f B0 ⊆ U . Now pN ≡ N −1 fg, where 0 ≤ g ≤ pN , as a projection belongs to B0 , therefore so does g. As a result, we obtain that pN ∈ N −1 f B0 ⊆ U. So for each non-zero projection p and each 0-neighbourhood U in A[τ ], we can find a projection pN in U , such that 0 < pN ≤ p. Moreover, for each m ≥ N, we find a projection pm ≡ N −1 f (gpm ) with pm ≤ pN and pN pm = pm . On the other hand, g, pm ∈ B0 , therefore gpm ∈ B02 ⊆ B0 . Thus, pm ≡ N −1 f (gpm ) ⊆ N −1 f B0 ⊆ U.

(4.4.14)

In a similar way, one can show that every decreasing sequence with null intersection converges to 0, with respect to τ . Now, let Qi , i = 1, 2, 3, . . ., be an enumeration of the open intervals in [0, 1], with rational end-points. Let the qi ’s be the associated to Qi ’s projections. Suppose that, for each i, we define a projection si ≤ qi , such that the corresponding Borel sets Si satisfy the inclusion Si ⊆ Qi \

i−1  j =1

Sj and Si = ∅, only if , Qi \

i−1  j =1

Sj = ∅,

102

4 Commutative Generalized B*-Algebras: Functional Calculus and Equivalent. . .

(always modulo sets of first category). Using the fact that every Borel set is congruent to  an open set, modulo sets of first category (see [100, § 11, III]), we obtain that ∞ j =1 Sj = [0, 1]. The Borel sets Si are disjoint by their choice, therefore n  j =1

sj → 1, as n → ∞; hence,

∞ 

sj = 1,

(4.4.15)

j =1

with respect to τ . We  show (4.4.15). The projections  si correspond to the Borel sets Si , so that 1 − nj=1 sj corresponds to [0, 1] \ ∞ j =1 Sj = ∅. It follows that

 n (1 − j =1 sj )n is a decreasing sequence of projections with null intersection, therefore it converges to 0, with respect to τ . Thus, (4.4.15) has been proven. Now, suppose that U is a closed 0-neighbourhood in A[τ ] that does not contain the identity 1. Since τ is a locally convex topology, we may find a sequence of 0-neighbourhoods in A[τ ], such that U = U0 ⊇ U1 ⊇ U2 ⊇ . . . with Uj + Uj ⊆ Uj −1 , ∀ j. According to (4.4.14), sj may be chosen, such that sj ∈ Uj , for all j ’s. But then, ∞ j =1 sj = 1 ∈ U , which is a contradiction. Thus, indeed not every ∗-algebra of functions admits a GB∗ -topology. Notes The results of Sects. 4.1–4.3 are due to G.R. Allan and for these the reader is referred to [5, Sections 3, 4 and 5]). The content of Sect. 4.4 consists of results that are owed to P.G. Dixon and can be found in [50].

Chapter 5

Extended C*-Algebras and Extended W*-Algebras

In this chapter, we discuss GB∗ -algebras of closable operators in a Hilbert space. Thus, the reader has the chance to encounter the basics of unbounded ∗-representations (Sect. 5.1), with various uniform topologies defined on the ∗preserving vector space L† (D, H) of all linear operators T with domain a dense subspace D of a Hilbert space H and values in H, in such a way that D is contained in the domain D(T ∗ ) of the adjoint operator T ∗ of T . The restriction of T ∗ on D defines an involution on L† (D, H), detoted by † (Sect. 5.2). Sections 5.3 and 5.4 deal with EC∗ -algebras and EW∗ -algebras that correspond to unbounded generalizations of the classical C ∗ - and W ∗ -algebras, respectively. The latter algebras occur naturally in the analysis of an unbounded generalization of left Hilbert algebras (see Sect. 5.4 and [84]).An interesting contribution of EC∗ -algebras, to the context of GB∗ -algebras, the reader can enjoy in Theorem 6.3.5. For more information about unbounded generalizations of von Neumann algebras, the reader is referred to [7] and [85].

5.1 O∗ -Algebras and Unbounded *-Representations O∗ -algebras play an important role in this chapter. So, first we present the definition of an O∗ -algebra and some basic facts on their theory. For more detail the reader is referred to [7, 85, 136, 138]; see also the Notes at the end of Sect. 5.1. Let D be a dense subspace of a Hilbert space H. For a given Hilbert space H, the symbol  , will stand for the inner product of H, while  ·  will denote the norm on H resulting from this inner product. Let L† (D) be the set of all linear operators T from D to D, such that the domain D(T ∗ ) of the adjoint T ∗ of T contains D and T ∗ D ⊂ D. Then, L† (D) is a ∗algebra under the usual algebraic operations and the involution given by the map T → T † := T ∗ D . © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Fragoulopoulou et al., Generalized B*-Algebras and Applications, Lecture Notes in Mathematics 2298, https://doi.org/10.1007/978-3-030-96433-7_5

103

104

5 Extended C*-Algebras and Extended W*-Algebras

In this regard, if T , S are two linear operators in H, we say that S is an extension of T and we write T ⊂ S, if D(T ) ⊂ D(S) and T ξ = Sξ , for every ξ ∈ D(T ). Furthermore, a linear operator T in H is said to be closed if its graph G(T ) is closed in H ⊕ H, while T is called closable if it has a closed extension. Every closable operator has a minimal closed extension, called closure of T , denoted by T . Definition 5.1.1 Let T be a closed operator in a Hilbert space H. A subspace D of the domain D(T ) of T is called a core for T if T  D = T . A densely defined linear operator T in H is said to be symmetric if T ⊂ T ∗ (i.e., ∗ T is an extension of T ). If T ∗ = T , then T is called self-adjoint . A symmetric operator T in H is called essentially self-adjoint if T = T ∗ , i.e., D(T ) is a core for T ∗ . A ∗-subalgebra M of L† (D) is called an O∗ -algebra on D in H. The locally convex topology on D induced by the seminorms    · T : T ∈ M , where ξ T := ξ  + T ξ , ξ ∈ D,

(5.1.1)

is called the graph topology on D and is denoted by tM . If the locally convex space  D[tM ] is complete, then M is called a closed O∗ - algebra. Denote by D(M) the completion of D[tM ]. Then,  D(M) =



D(T ),

T ∈M

where T is the closure of the operator T . Put now,   := T    T D(M) , T ∈ M and M := {T : T ∈ M}.  is the smallest closed extension of M, which is called the closure of M. Then, M It is easily shown that  ⇔ D = D(M)  M is closed ⇔ M = M =



D(T ).

T ∈M

Let M bean O∗ -algebra on D in H. Then, M is said to be self-adjoint if := T ∈M D(T ∗ ) = D. It is clear that if M is self-adjoint, then it is closed. A closed O∗ -algebra M is said to be integrable if T ∗ = T † , for every T ∈ M. For integrable O∗ -algebras we have the following, for the proof of which we refer to Theorem 7.1 in [125] and in [78]. D∗ (M)

Proposition 5.1.2 Let M be a closed O∗ -algebra on D in H. The following are equivalent: (i) M is integrable.

5.1 O ∗ -Algebras and Unbounded *-Representations

105

(ii) T is a self-adjoint operator, for each T in the set H (M) := {T ∈ M : T † = T }. Thus, if one of the above equivalent statements (i) and (ii) holds, then M is selfadjoint and M := {T : T ∈ M} is a ∗-algebra of closed operators in H equipped with the strong sum T + S := T + S = T + S, the strong scalar multiplication α · T :=

 αT ,

if α = 0

0,

if α = 0

=

αT , α ∈ C,

the strong product T · S := T S = T S and the involution T → T ∗ := T † . Now, let A be a ∗-algebra and D be a dense subspace of a Hilbert space H. A ∗-representation of A on D is a ∗-homomorhism π from A into L† (D), such that π(e) = I , when A has an identity e and I is the identity operator on D. The space D is called the domain of π and it will be denoted by D(π). The respective Hilbert space, in which the domain of π is dense, is accordingly denoted by Hπ [136, p. 38]. A ∗-representation π of A is said to be faithful if it is injective. If two ∗-representations π1 , π2 of A are given, we write π1 (a) ⊂ π2 (a), a ∈ A, if D(π1 ) ⊂ D(π2 ) and π1 (a)ξ = π2 (a)ξ , for every ξ ∈ D(π1 ). If π1 (a) ⊂ π2 (a), for all a ∈ A, we say that π2 is an extension of π1 and we denote it by π1 ⊂ π2 . Let π be a ∗-representation of a ∗-algebra A. Denote by tπ the graph topology tπ(A) on D(π) with respect to the O∗ -algebra π(A). If D(π)[tπ ] is complete, i.e.,  the O∗ -algebra π(A) is closed, then π is said to be closed. Denote by D(π) the completion of D(π)[tπ ] and put  π (x) ≡ π(x) D (π) , x ∈ A. Then  π is a closed ∗-representation of A, which is the smallest closed extension of π and it is called the closure of π. A closed ∗-representation π of a ∗-algebra A is said to be self-adjoint (resp. integrable), if the O∗ -algebra π(A) is self-adjoint (resp. integrable). Example 5.1.3 Let D be a dense subspace of a Hilbert space H. (1) Let T be a self-adjoint element of L† (D), that is T † = T and P(T ) be the  ) of the O∗ -algebra polynomial algebra generated by T . Then, the closure P(T  ) is self-adjoint, if and only if, T n is P(T ) is integrable, if and only if, P(T essentially self-adjoint, for every n ∈ N. (2) Let S and T be self-adjoint elements in L† (D), such that ST = T S. Let P(S, T ) be the commutative O∗ -algebra on D generated by S and T . Then, the closure  T ) of the O∗ -algebra P(S, T ) is integrable, if and only if, there exists a P(S, normal operator, which is an extension of the operator S + iT .

106

5 Extended C*-Algebras and Extended W*-Algebras

Example 5.1.4 Let H := L2 (R) and D := C0∞ (R), the space of all compactly supported smooth functions on R. Define M, as follows: M :=

" m  n k=0 l=0

αkl t k (

# d l ) : αkl ∈ C, n, m ∈ N ∪ {0} . dt

 Then, D(M) coincides with the Schwartz space S(R) of all rapidly decreasing  (as in the discussion after (5.1.1)) is a closed O∗ smooth functions on R and M algebra on D in H. Let A be the CCR-algebra for one degree of freedom (see, for instance, [53, p. 69, Definition 4.10]), that is the ∗-algebra generated by the identity e and two self-adjoint elements p, q satisfying the Heisenberg commutation relation pq − qp = −ie. The ∗-representation π of A on S(R) in the Hilbert space L2 (R), defined by



 df π(e) = I, π(p)f (t) = tf (t), π(q)f (t) = −i , f ∈ S(R), t ∈ R, dt  = π(A), so M  is not is called Schrödinger representation. Then, we have M integrable. Notes Definition 5.1.1 is found in [7, pp. 9, 11]. For the Example 5.1.3(1) see [89, Theorem 2.1], for (2) of the same example see [89, Theorem 3.2], while for the Example 5.1.4 cf. [83, Example 1.1.5]. The original examples of O∗ -algebras were described by Borchers (1962) [33] and Uhlmann (1962) [147], who studied examples of O∗ -algebras, called Borchers algebras, arising from the Wightman axioms of quantum field theory. Powers (1971) [125] and Lassner (1972) [103] began a systematic study of algebras of unbounded operators, whose O∗ -algebras are an important subclass.

5.2 Uniform Topologies on O∗ -Algebras Let D be a dense subspace of a Hilbert space H and M a closed O∗ -algebra on D in H. We shall introduce on M the weak, strong and strong∗ topology (see [85, p. 23, 1.6(A)]). Consider the set L† (D, H) of all linear operators T from D to H, such that D is contained in the domain D(T ∗ ) of the adjoint operator T ∗ of T . Then, L† (D, H) becomes a ∗-preserving vector space endowed with the usual addition and scalarmultiplication and with the involution T † := T ∗ D , T ∈ L† (D, H). Consider the

5.2 Uniform Topologies on O ∗ -Algebras

107

following families of seminorms: pξ,η (T ) := |T ξ, η |, T ∈ L† (D, H) and ξ, η ∈ D, pξ (T ) := T ξ , T ∈ L† (D, H) and ξ ∈ D,

(5.2.1)

ξ

p∗ (T ) := pξ (T ) + pξ (T † ), T ∈ L† (D, H) and ξ ∈ D. The topologies induced by the families of seminorms 

    ξ  pξ,η : ξ, η ∈ D , pξ : ξ ∈ D and p∗ : ξ ∈ D

are called the weak, strong and strong∗ topology on L† (D, H), respectively and they will be denoted, respectively by τw , τs and τs∗ . The locally convex spaces corresponding to the preceding topologies will be denoted by L† (D, H)[τw ], L† (D, H)[τs ] and L† (D, H)[τs∗ ], respectively. The latter locally convex space is complete. The restrictions of the topologies τw , τs and τs∗ on M are called weak, strong and strong∗ topology on M. It is easy to see that M[τw ] is a locally convex ∗-algebra, but M[τs ] and M[τs∗ ] are not necessarily locally convex algebras. This happens, for instance, with L† (D)[τs ] and L† (D)[τs∗ ]. For this reason, we define the following quasi-strong (resp. quasi-strong∗) topology on M, under which M becomes a locally convex (resp. a locally convex ∗-)algebra. Let ξ ∈ D and T ∈ L† (D). We put ξ

pT (S) := T Sξ , ∀ S ∈ M, ξ

p∗,T (S) := T Sξ  + T S † ξ , ∀ S ∈ M.

(5.2.2)

Each of the preceding functions is a seminorm on M. Let A be an O∗ -algebra on D containing the O∗ -algebra MI on D generated by M and the identity operator I . We call the locally convex topology on M induced by the family of seminorms  ξ  A. pT , T ∈ A, ξ ∈ D quasi-strong topology on M for A and we denote it by τqs  ξ The locally convex topology on M defined by the family of seminorms p∗,T , T ∈  A, ξ ∈ D is called quasi-strong∗ topology on M for A and we denote it by L† (D )

∗,A . Such topologies depend, in general, on A. In particular, the topology τqs τqs

∗,L (D ) MI ∗,MI (resp. τqs ) is the strongest and τqs (resp. τqs ) the weakest among such quasi-strong (resp. quasi-strong∗) topologies. Now note that for any T ∈ A and ξ ∈ D, we have †

ξ

S ξ

ξ

pT (S1 S2 ) = T S1 S2 ξ  = pT2 (S1 ) = pT S1 (S2 ), ∀ S1 , S2 ∈ M. A ] is a locally Since S2 ξ ∈ D and T S1 ∈ A (as M ⊂ A), it follows that M[τqs ∗,A ] is a locally convex ∗-algebra. convex algebra and M[τqs

108

5 Extended C*-Algebras and Extended W*-Algebras

 The quasi-strong topology

L† (D )

τqs

and

the quasi-strong∗ topology

∗,L† (D )

∗, τqs , for simplicity’s sake, from now on will be denoted by τqs , τqs respectively.

The following diagram shows the relationship of all the foregoing locally convex topologies defined on M in this section: τw ≺ τs ≺ τs∗ ≺

≺

MI ∗,MI τqs ≺ τqs

(5.2.3)

≺

≺

∗ τqs ≺ τqs .

We next define various uniform topologies on a closed O∗ -algebra M on D. Let M denote the set of all tL† (D)-bounded subsets of D. Then, for any M ∈ M we define: pM (T ) := sup |T ξ, η |, ∀ T ∈ L† (D), ξ,η∈M

p (T ) := sup T ξ , ∀ T ∈ L† (D), M

ξ ∈M

(5.2.4)

p∗M (T ) := pM (T ) + pM (T † ), ∀ T ∈ L† (D).       The families of seminorms pM : M ∈ M , pM : M ∈ M and p∗M : M ∈ M define locally convex topologies on L† (D), denoted respectively by τu , τ u , τ∗u and called uniform topologies. Since M is closed, it follows from [136, Corollary 2.3.11] that a subset M of D is tL† (D) -bounded (i.e., M ∈ M), if and only if, M is tM -bounded. Therefore, the induced locally convex topologies on M by the uniform topologies τu , τ u and τ∗u of L† (D), will be also denoted by the preceding symbols. Using the very definitions, it follows easily that M[τu ] is a locally convex ∗-algebra, but M[τ u ] and M[τ∗u ] are not necessarily locally convex algebras. Conditions under which M[τ u ] is a locally convex algebra and M[τ∗u ] is a locally convex ∗-algebra can be found in Proposition 5.2.1(2)(3) and Corollary 5.2.2, below. For this, as in the case of the quasi-strong and quasi-strong∗ topologies, we introduce some extra uniform topologies on M. Let T ∈ L† (D) and M ∈ M

5.2 Uniform Topologies on O ∗ -Algebras

109

be arbitrary. Define pTM (S) := pM (T S) = sup T Sξ , ∀ S ∈ M, ξ ∈M

M (S) p∗,T

:=

pTM (S)

(5.2.5)

+ pTM (S † ),

∀ S ∈ M.

Let A be an O∗ -algebra induced by the  on D containing MI . The  topologies  M families of seminorms pTM : T ∈ A, M ∈ M and p∗,T : T ∈ A, M ∈  M , respectively, are called quasi-uniform and quasi-uniform∗ topologies on M A and τ ∗,A , respectively. Then it is shown that for A and will be denoted by τqu qu

A ] is a locally convex algebra and M[τ ∗,A ] a locally convex ∗-algebra. Such M[τqu qu L† (D )

∗,L† (D )

topologies depend on A. In particular, τqu (resp. τqu ) is the strongest MI ∗,MI topology and τqu (resp. τqu ) the weakest topology among such quasi-uniform L† (D )

∗,L† (D )

and τqu the quasi-uniform and the quasitopologies. Here we call τqu ∗ , respectively. uniform∗ topology and denote them by τqu and τqu It is clear that when M is a ∗-algebra of bounded operators, the uniform and quasi-uniform topologies coincide with the usual uniform topology on M. In the unbounded case, this does not happen, in general, as is shown below. Proposition 5.2.1 (1) τu ≺ τ u ≺ τ∗u ≺

≺

MI ∗,MI τqu ≺ τqu

≺

≺

∗ τqu ≺ τqu .

MI (2) M[τ u ] is a locally convex algebra, if and only if, τ u = τqu . ∗,MI (3) M[τ∗u ] is a locally convex ∗-algebra, if and only if, τ∗u = τqu . MI (4) If tM = tL† (D), in particular, if D[tM ] is a Fréchet space, then τqu = τqu

∗ = τ ∗,MI . and τqu qu † ∗ ] is complete. (5) L (D)[τqu

Proof (1) is easily shown by the very definitions. (2) Suppose that M[τ u ] is a locally convex algebra. Then, for any S ∈ M, the map MI T ∈ M[τ u ] → ST ∈ M[τ u ] is continuous, which implies that τqu ≺ τ u. MI The converse follows, since M[τqu ] is a locally convex algebra.

110

5 Extended C*-Algebras and Extended W*-Algebras

(3) is shown in a similar way as (2). (4) Take arbitrary T ∈ L† (D) and M ∈ M. Since tM = tL† (D) there exists an operator K ∈ M such that T ξ  ≤ Kξ , for all ξ ∈ D. Hence, M M M pTM (S) ≤ pK (S) and p∗,T (S) ≤ p∗,K (S), ∀ S ∈ M,

so that, taking also into account the corresponding definitions, we conclude that MI ∗ = τ ∗,MI . and τqu τqu = τqu qu Suppose now that D[tM ] is a Fréchet space. Then, by the closed graph theorem, every operator T ∈ L† (D) is continuous from the Fréchet space D[tM ] into the Hilbert space H. Thus, we obtain tM = tL† (D) , hence the conclusion. ∗ ]. Hence, for every S ∈ (5) Take an arbitrary Cauchy net (Tν )ν∈ in L† (D)[τqu L† (D) and every ξ ∈ D, the sequence (STν ξ )ν∈ converges strongly in ∗ ' τ ∗ . It follows that lim ST ξ exists in H. Put ST ξ := H, since τqu ν ν s limν STν ξ, ξ ∈ D. Then, for T ∈ L† (D, H), we have that |S ∗ η, T ξ | = lim |η, STν ξ | ≤ η  lim STν ξ , ν

ν

for each S ∈ L† (D) and η ∈ D(S ∗ ).Therefore, since L† (D) is closed by the closedness of M, we have T ξ ∈ S∈L†(D) D(S) = D, from which we M (T − conclude that T ∈ L† (D). Furthermore, it can be shown that limν p∗,S ν † † ∗ T ) = 0, for all S ∈ L (D) and M ∈ M. Thus, L (D)[τqu ] is complete. For a more detailed proof the reader is referred to [136, Proposition 3.3.15].   The next corollary provides another condition under which M[τ u ] is a locally convex algebra and M[τ∗u ] is a locally convex ∗-algebra. Corollary 5.2.2 Suppose that there exists a subset C of L† (D), such that ST = T S, for all T ∈ M and S ∈ C and tM = tC (take, for example, M to be commutative). Then, M[τ u ] is a locally convex algebra and M[τ∗u ] is a locally convex ∗-algebra. Proof Take an arbitrary operator T ∈ M and an arbitrary set M from M. Since tM = tC , there exists a finite subset {K1 , K2 , . . . , Kn } of C, such that T ξ  ≤

n 

Kk ξ , ∀ ξ ∈ D.

k=1

This implies that pTM (S) ≤

n  k=1

M pKk M (S) and p∗,T (S) ≤

n  k=1

p∗Kk M (S), ∀ S ∈ M.

5.2 Uniform Topologies on O ∗ -Algebras

111

∗ . Since Kk M ∈ M, for k = 1, 2, . . . , n, we conclude that τ u = τqu and τ∗u = τqu Applying now Proposition 5.2.1(1)(2)(3), we have the conclusion.  

Further, we study conditions under which all uniform and all quasi uniform topologies are identical. Proposition 5.2.3 Let M be a closed O∗ -algebra on D in H. The following statements are equivalent: (i) M[τu ] has jointly continuous multiplication; (ii) τu = τ u ; ∗ . (iii) τu = τqu Proof (i) ⇒ (iii) Take arbitrary M ∈ M and T ∈ L† (D). Then by (i) there is an element M1 ∈ M (see also (5.2.4), (5.2.5)), such that pTM (S)2 = pM (S † T † T S) ≤ pM1 (T S)2 = sup |Sξ, T † η |2 ξ,η∈M1

≤

sup ξ,η∈M1 ∪T † M1

|Sξ, η |2

= pM1 ∪T † M1 (S)2 , ∀ S ∈ M. Since M1 ∪ T † M1 ∈ M, we obtain that τu = τqu . But the involution T → T † is ∗ . τu -continuous, therefore using (5.2.5) we conclude that τu = τqu = τqu (iii) ⇒ (ii) Follows from (iii) and Proposition 5.2.1(1). (ii) ⇒ (i) Let M ∈ M. Then, pM (T S) = sup |Sξ, T † η | ≤ sup T † ηSξ  ξ,η∈M

ξ,η∈M

= pM (T † )pM (S), ∀ T , S ∈ M. The assertion now follows from (ii) and the τu -continuity of the involution †. This completes the proof.   The example that follows, provides a closed O∗ -algebra, on which all uniform and all quasi uniform topologies coincide. 

Example 5.2.4 Let Mn n∈N be a sequence of infinite dimensional ∗-algebras of bounded operators acting on the Hilbert spaces Hn , n ∈ N. Suppose that each Mn contains the identity operator. Denote by H the Hilbert space (orthogonal) direct sum of the Hilbert spaces Hn , n ∈ N and let  D := ξ = (ξn ) ∈ n∈N Hn : ξn = 0, ∀ n ∈ N,  except for a finite number of them .

112

5 Extended C*-Algebras and Extended W*-Algebras

Then, the product M := n∈N Mn is a closed O∗ -algebra on D in H, under the following algebraic operations: T +S := (Tn + Sn ), λT := (λTn ), T S := (Tn Sn ), T † := (Tn∗ ), where T = (Tn ), S = (Sn ) in M and λ ∈ C. It is shown that for any M ∈ M there exists a positive integer N such that   M ⊆ ξ = (ξn ) ∈ D : ξn = 0, ∀ n > N . This implies that τu = τ u ; therefore Proposition 5.2.3 gives that all uniform and all quasi uniform topologies are equal on M. Furthermore, all previous topologies are also equal to the locally uniform topology τlu defined by the family { · n : n ∈ N} of seminorms, where T n := Tn  (the operator norm of Tn ), for all T = (Tn ) ∈ M. For more details the reader is referred to [87, Proposition 3.5]. Example 5.2.5 Let D be a dense subspace of a Hilbert space H and P(T ) the O∗ algebra of all polynomials generated by an unbounded operator T in L† (D). Let

∞ be the set of all positive sequences (rn ) with 1 ≤ r0 ≤ r1 ≤ . . .. We introduce the topology τ∞ defined by all the seminorms p(T )(rn ) :=

∞  n=0

rn |αn |, (rn ) ∈ ∞ and p(T ) =

k 

αn T n ∈ P(T ).

n=0

Then, τ∞ is the finest locally convex topology on the O∗ -algebra P(T ). Suppose that P(T ) is closed. Then, we have that ∗ τs = τs∗ = τu = τ u = τ∗u = τqu = τ∞ .

We next prove that τu = τ u , for every maximal O∗ -algebra L† (D). Example 5.2.6 Let D be a dense proper subspace of a Hilbert space H. Suppose that L† (D) is closed. Then, we have the following: (1) The multiplication of L† (D) is not jointly continuous, under any of the uniform and quasi uniform topologies. ∗ . (2) τu = τ u , τ u = τqu and τ∗u = τqu † † u (3) L (D)[τu ] and L (D)[τ∗ ] are not complete. Indeed, suppose that multiplication in L† (D)[τqu ] is jointly continuous. Take an arbitrary η ∈ D with η = 1. Then, for M = {η} ∈ M, there exists T ∈ L† (D)

5.2 Uniform Topologies on O ∗ -Algebras

113

and M1 ∈ M such that pM (S(ξ ⊗ η)) = Sξ  ≤ pTM1 (S)pTM1 (ξ ⊗ η) = pTM1 (S) sup ζ, η T ξ  ζ ∈M1

≤ pTM1 (S)( sup ζ )T ξ , ∀ S ∈ L† (D) and ξ ∈ D, ζ ∈M1

where ξ ⊗ η is the operator ζ → ζ, η ξ, ζ ∈ H. Thus, D(T ) =



D(S) = D = D(T ),

S∈L† (D )

therefore T is closed. Hence, D becomes a Hilbert space, denoted by HT , under the inner product ξ, η

T

= T ξ, T η , ∀ ξ, η ∈ D.

Now, for any x ∈ H, we put f (ξ ) := T ξ, x , ∀ ξ ∈ HT . Then, f is a continuous linear functional on the Hilbert space HT , so that by the Riesz representation theorem there exists an element η ∈ HT , such that f (ξ ) := T ξ, x = T ξ, T η = ξ, T † T η , ∀ ξ ∈ D = HT . It follows that x ∈ D(T ∗ ), hence D(T ∗ ) = H. By the closed graph theorem T ∗ is then a bounded operator, therefore T = T ∗∗ is a bounded operator on H. Hence, D = H, which is a contradiction according to our hypothesis. So, the multiplication of L† (D)[τqu ] is not jointly continuous. In the same way, we can show that the multiplication is not jointly continuous if we consider any of the other uniform and quasi uniform topologies on L† (D). Thus, (1) is proved. Now, by Proposition 5.2.3 and (1) we obtain that τu = τ u . We show that τ∗u = ∗ τqu . Suppose we have equality. Take arbitrary T ∈ L† (D) and η ∈ D with η = 1. Then, for M = {η} ∈ M there exists M1 ∈ M, such that pTM (ξ ⊗ η) = T ξ  ≤ pM1 (ξ ⊗ η) + pM1 ((ξ ⊗ η)† ) ≤ 2( sup ζ )ξ , ∀ ξ ∈ D. ζ ∈M

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5 Extended C*-Algebras and Extended W*-Algebras

 Hence, T is bounded. Consequently, D = T ∈L† (D ) D(T ) = H, which is a ∗ . contradiction. Thus, τ∗u = τqu We show now (3). Let x ∈ H\D. Then, for any M ∈ M and ξ ∈ D we have 

2

pM (ξ ⊗ ξn ) − (ξ ⊗ x) ≤ sup ζ  ξ ξn − x and

M



ζ ∈M1



 p∗ (ξ ⊗ ξn ) − (ξ ⊗ x) ≤ 2 sup ζ  ξ ξn − x, ζ ∈M1

where (ξn ) is a sequence in D converging to x. Hence, (ξ ⊗ξn ) is a Cauchy sequence in both L† (D)[τu ] and L† (D)[τ∗u ], but has no limit in any of them. This completes the proof of (3). Notes The Example 5.2.6 is Proposition 3.3.19 in [136]. The uniform topologies ∗ were first defined by τu , τ u , τ∗u and the quasi uniform topologies τqu , τqu G. Lassner in [103]. The preceding notations, used in this section, are different from those of Lassner. There are another two uniform topologies, called ρ-topology and λ-topology, introduced by J.P. Jurzuk (see, [92–94]), but we do not treat these topologies here. The example 5.2.5 is due to K. Schmüdgen [134]; see also [87, Proposition 3.8]. For almost all the results in this section, we refer the reader to [136] and [87].

5.3 Extended C*-Algebras In this section, we investigate conditions under which the locally convex algebra M[τu ] of a closed O∗ -algebra M, becomes a (locally convex) GB∗ -algebra of Dixon (Definition 3.3.5). We first introduce the so-called extended C ∗ -algebras. Throughout this section, let M be a closed O∗ -algebra on D in H, containing the identity operator I, where D is a dense subspace of a Hilbert  space H. Let Mb bethe bounded part of a closed O∗ -algebra M, that is, Mb := T ∈ M : T ∈ B(H) . Definition 5.3.1 A closed O∗ -algebra M is called symmetric if (I + T † T )−1 belongs to Mb , for all T ∈ M. If M is symmetric and Mb := T : T ∈ Mb is a C ∗ -algebra (resp. a von Neumann algebra), then M is said to be an EC∗ -algebra (abbreviation of extended C ∗ -algebra), respectively an EW∗ -algebra (abbreviation of ‘extended W ∗ -algebra’) on D over Mb . By Theorem 2.3 in [78] we have the following Proposition 5.3.2 A closed symmetric O∗ -algebra M on D in H, in particular,  an EC∗ -algebra M is integrable. By Proposition 5.1.2, M := T : T ∈ M is a ∗-algebra of closed operators in H equipped with the strong sum, the strong scalar multiplication, the strong product and the involution T → T † .

5.3 Extended C*-Algebras

115

Consider now a closed O∗ -algebra on D in H and let τ be one of the topologies ∗ ∗,M ∗ ∗,M τw , τqs , τqs , τu , τqu , τqu

on M. Then, by the discussion after (5.2.1), as well as by (5.2.3) and Proposition 5.2.1(3), M[τ ] is a locally convex ∗-algebra and τw ≺ τ . Recall that B∗M[τ ] is the collection of all closed, bounded and absolutely convex subsets B of M[τ ], such that (see Definition 3.3.1) I ∈ B, B 2 ⊂ B and B ∗ = B. In this regard, we have the following   Lemma 5.3.3 The set U (M) := T ∈ Mb : T  ≤ 1 is a greatest member in B∗M[τ ] . Proof It is easily shown that U (M) ∈ B∗M[τ ] . We still show that U (M) is a greatest element in B∗M[τ ] . Take arbitrary B ∈ B∗M[τ ] . Suppose that there exists S ∈ B such that S > 1. Then, there is an element ξ ∈ D with ξ  = 1 and Sξ  > 1, therefore ! ! n n+1 lim !(S † S)2 ξ, ξ ! ≥ lim Sξ 2 = ∞.

n→∞

n→∞

On the other hand, since τw ≺ τ and B is τ -bounded, it will ! too. ! be τw -bounded n n Moreover, (S † S)2 ∈ B, for every n ∈ N. Hence, limn→∞ !(S † S)2 ξ, ξ ! < ∞, which is a contradiction. Therefore, B ⊆ U (M) and this completes the proof.   Lemma 5.3.3 implies the following Theorem 5.3.4 Let M be a closed O∗ -algebra. Then, M is an EC∗ -algebra, if and only if, M[τ ] is a (locally convex) GB∗ -algebra of Dixon, where τ is one of the ∗ , τ ∗,M , τ , τ ∗ , τ ∗,M on M. topologies τw , τqs u qs qu qu Proof It is immediate from Definition 5.3.1, and Lemma 5.3.3.

 

For another characterization of a certain closed O∗ -algebra, as an EC∗ -algebra, in terms of the set of ‘commutatively quasi-positive’ elements of a ‘locally convex quasi C ∗ -algebra’, see [13, Proposition 4.5]. An O∗ -algebra is said to be C ∗ -like if it is a C ∗ -like locally convex ∗-algebra (see Definition 3.5.1). Proposition 5.3.5 Every C ∗ -like O∗ -algebra is an EC∗ -algebra. Proof Let M be a C ∗ -like O∗ -algebra on D in H with a C ∗ -like family = {pν }ν∈ of seminorms. By Theorem 3.5.3 M is a GB∗ -algebra, whose closed unit ball U (p ) is the greatest element in B∗A . For any ξ ∈ D we can define a positive linear functional ωξ on the C ∗ -algebra D(p ) by ωξ (T ) = T ξ, ξ , T ∈ D(p ). Then, T ξ  ≤ p (T )ξ , for each T ∈ D(p ) and ξ ∈ D. It follows that

116

5 Extended C*-Algebras and Extended W*-Algebras

D(p ) ⊂ Mb and p equals the operator norm. Furthermore, we can prove as in the proof of Theorem 3.5.3 that D(p ) is dense in Mb with respect to the operator norm. Therefore, it turns out that D(p ) = Mb , consequently M is an EC∗ -algebra. This completes the proof.   By Theorems 3.5.3 and 6.3.5 (in Sect. 6.3) we have the following Theorem 5.3.6 Every C ∗ -like locally convex ∗-algebra is isomorphic to a C ∗ -like O∗ -algebra. Notes P.G. Dixon defined EC∗ -algebras and EW∗ -algebras of closed operators in a Hilbert space, in [48, 49]. But, it is difficult to introduce a locally convex topology on EC∗ -algebras, in general. From this viewpoint, in this chapter, we dealt with EC∗ -algebras and EW∗ -algebras of closable operators on a common dense subspace in a Hilbert space, defined by A. Inoue in [78]. For the results of this section the reader is referred to [78, 87]. Proposition 5.3.5 and Theorem 5.3.6 are found in [88]. For a study of irreducible representations of EC∗ -algebras and EW∗ -algebras, the reader may consult [21].

5.4 Left Extended W*-Algebras of Unbounded Hilbert Algebras It is well known that semifinite von Neumann algebras are related to Hilbert algebras; for the latter term, see [46, Chapter 5]. That is, if A0 is a Hilbert algebra, then the left von Neumann algebra U0 (A0 ) is defined and it is a semifinite von Neumann algebra. Conversely, if M0 is a semifinite von Neumann algebra, then it is isomorphic to the left von Neumann algebra U0 (A0 ) of a Hilbert algebra A0 (see [46, Part I, Chapter 6, Theorem 2]). In this section, we extend the preceding results to EW∗ -algebras (see Definition 5.3.1). Our starting point will be an unbounded extension of Hilbert algebras. Let A be a ∗-algebra, which is also a pre-Hilbert space with inner product  , . Let H be the completion of A with respect to the norm induced by the inner product. Suppose that A satisfies the following properties: (i) ξ, η = η∗ , ξ ∗ , ∀ ξ, η ∈ A; (ii) ξ η, ζ

= η, ξ ∗ ζ , ∀ ξ, η, ζ ∈ A.

Now we define the operators π(ξ ) and π (ξ ), ξ ∈ A, by π(ξ )(η) := ξ η and π (ξ )(η) := ηξ, ∀ η ∈ A.

5.4 Left Extended W*-Algebras of Unbounded Hilbert Algebras

117

Then, π is a ∗-representation of A in H with domain D(π) = A and π(ξ ∗ ) ⊂ π(ξ )∗ , for every ξ ∈ A, while π is a linear map of A into L† (A), such that π (ξ ∗ ) ⊂ π (ξ )∗ and π (ξ ξ1 ) = π (ξ1 )π (ξ ), ∀ ξ, ξ1 ∈ A. Such a π is called a ∗-antirepresentation. Definition 5.4.1 If A satisfies the properties (i), (ii) and   (iii) A20 is dense in H, where A0 := ξ ∈ A : π(ξ ) ∈ B(H) , then A is called an unbounded Hilbert algebra over A0 in H. In particular, if A0 = A, then A is said to be pure. If A = A0 , then A is named a Hilbert algebra in H. The closure of the ∗-representation π (resp. π ) is called the left (resp. right) regular

). representation of A and it is denoted by πA (resp. πA For the left and right regular representation of an unbounded Hilbert algebra, we have the following Theorem 5.4.2 Let A be an unbounded Hilbert algebra over A0 in H. Then, the left

of A are integrable. Hence, π (A) is ∗-algebra regular representations πA and πA A of closed operators in H under the algebraic operations of strong sum, strong scalar multiplication, strong product and the involution given by † (see, Proposition 5.3.2). Proof It is clear that A0 is a Hilbert algebra in H. Let π0 (resp. π0 ) be the left (resp. right) regular representation of the Hilbert algebra A0 . More precisely, for any x ∈ H we define π0 (x) and π0 (x) by π0 (x)ξ := π0 (ξ )x and π0 (x)ξ := π0 (ξ )x, ∀ ξ ∈ A0 . Then, π0 (x) and π0 (x) are linear operators in H with domain A0 . The involution on A is extended to an involution on H, which is also denoted by ∗. Then, for ξ ∈ A, by the definition of πA and π0 we have πA (ξ )η = π0 (η)ξ = π0 (ξ )η, ∀ η ∈ A0 , so that π0 (ξ ) ⊂ πA (ξ ). Furthermore, from [119, Theorem 3], it follows that π0 (x)∗ = π0 (x ∗ ) and π0 (x)∗ = π0 (x ∗ ), x ∈ H, which implies that πA (ξ )∗ ⊂ π0 (ξ )∗ = π0 (ξ ∗ ) ⊂ πA (ξ ∗ ) ⊂ πA (ξ )∗ , ξ ∈ A.

(5.4.1)

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5 Extended C*-Algebras and Extended W*-Algebras

Hence, we have πA (ξ )∗ = πA (ξ ∗ ) = πA (ξ )† , ξ ∈ A,

(5.4.2)

is integrable. which means that πA is integrable. Similarly, we can show that πA This completes the proof.  

In the sequel, we shall construct GB∗ -algebras through unbounded Hilbert algebras. The first step for this is the construction of an Lω -space by using the noncommutative integration theory of I.E. Segal [140]. The reader is referred to [117, 119, 140] for noncommutative integration of semifinite von Neumann algebras. Let U0 (A0 ) (resp. V0 (A0 )) be the left (resp. right) von Neumann algebra of a given Hilbert algebra A0 . It is well known that both are semifinite. Let ϕ0 be the natural trace on U0 (A0 )+ . If   (A0 )b := x ∈ H : π0 (x) ∈ B(H) , then (A0 )b is a Hilbert algebra containing A0 . When A0 = (A0 )b , then A0 is called a maximal (or achieved) Hilbert algebra in H. Let M[U0 (M0 )] be the set of all measurable operators in H (in the sense of I.G. Segal) with respect to the von Neumann algebra U0 (A0 ) and M[U0 (A0 )]+ the set of all positive self-adjoint operators in M[U0 (A0 )]. For any T ∈ M[U0 (A0 )]+ we put   μ0 (T ) := sup ϕ0 (π0 (ξ )) : 0 ≤ π0 (ξ ) ≤ T , ξ ∈ (A0 )2b and   Lp (ϕ0 ) := T ∈ M[U0 (A0 )] : T p := μ0 (|T |p )1/p < ∞ , 1 ≤ p < ∞. Then, T p is called Lp -norm of T in Lp (ϕ0 ) and μ0 is called integral on Lp (ϕ0 ). If p = ∞, we identify L∞ (ϕ0 ) with U0 (A0 ) and denote by T  or T ∞ , the operator norm of T ∈ U0 (A0 ). It is well known that Lp (ϕ0 )[ · p ] is a Banach space for 1 ≤ p ≤ ∞. Now, for 2 ≤ p ≤ ∞, we define     p L2 (A0 ) := x ∈ H : π0 (x) ∈ Lp (ϕ0 ) and xp∧2 := max xp , x2 , p

x ∈ L2 (A0 ), where xp := π0 (x)p . In this regard, we have the next result

5.4 Left Extended W*-Algebras of Unbounded Hilbert Algebras

119

Lemma 5.4.3 The following hold: p

(1) L2 (A0 )[ · p∧2 ] is a Banach space and H = L22 (A0 ) ⊇ L2 (A0 ) ⊇ L2 (A0 ) ⊇ Lω2 (A0 ) ⊇ L∞ 2 (A0 ) = (A0 )b ,  p L2 (A0 ). 2 ≤ p < q, where Lω2 (A0 ) := p

q

2≤p 0, such that {(λx)n : n ∈ N} is a bounded subset of Lω2 (A0 )[τ2ω ]. Let π0 (x) = U |π0 (x)| be the polar decomposition of π0 (x). It is easily shown that {(λU ∗ x)n : n ∈ N} is a bounded subset of Lω2 (A0 )[τ2ω ]. Suppose that π0 (x) is unbounded. Then, there exists an element ξ ∈ (A0 )b , such that ξ  = 1 and π0 (y)ξ  > 1, where y := λU ∗ x. Hence, lim

n→∞

$

2n % n+1 π0 (y) ξ, ξ ≥ lim π0 (y)2 = ∞, n→∞

ω ω n which contradicts

ωthe factωthat  {y : n ∈ N} is a bounded

ω subset ωof L2 (A0 )[τ2 ]. Consequently, L2 (A0 )[τ2 ] 0 ⊆ (A0 )b . Thus, we have L2 (A0 )[τ2 ] 0 = (A0 )b .

126

5 Extended C*-Algebras and Extended W*-Algebras τω

We next show that (A0 )b 2 = Lω2 (A0 ). Let x be any element of Lω2 (A0 ). Let π  0∞(x) = U |π0 (x)| be the polar decomposition of the operator π0 (x) and |π0 (x)| = 0 λdEx (λ) be the spectral resolution of |π0 (x)|. Since,  Sn :=

n

λdEx (λ) ∈ U0 (A0 )



L2 (ϕ0 ), n ∈ N,

0

there exists a sequence (ξn )n∈N in (A0 )b , such that Sn = π0 (ξn ), n ∈ N. Then, for p≥2 U ξn − xp =  U π0 (ξn ) − U |π0 (x)| p ≤  π0 (ξn ) − |π0 (x)| p = −



∞

λp dϕ0 (Ex (λ)⊥ )

n

1/p

→ 0

n→∞

and U ξn ∈ (A0 )b , since U π0 (ξn ) = π0 (U ξn ), n ∈ N. Hence, it follows that (A0 )b is dense in Lω2 (A0 )[τ2ω ]. Suppose now that A0 has an identity element e. Then,it is easily shown that



 (A0 )b = U0 (A0 )e. Moreover, the set B0 ≡ U U0 (A0 ) e := Ae : A ∈ U U0 (A0 ) is a bounded absolutely convex subset of Lω2 (A0 )[τ2ω ], such that B02 ⊆ B0 and B0∗ = B0 . Furthermore, since B0 is τω -closed and τω ≺ τ2ω , it follows that B0 is τ2ω closed, therefore B0 ∈ B∗Lω (A0 ) . Thus, by Theorem 7.1.2 (in Sect. 7.1), we conclude 2 that Lω2 (A0 )[τ2ω ] is a GB∗ -algebra over U (U0 (A0 ))e.   An immediate consequence of Theorem 5.4.10 is the following Corollary 5.4.11 Let A be a pure unbounded Hilbert algebra over A0 . Then, A is a ∗-subalgebra of the Fréchet GB∗ -algebra Lω2 (A0 )[τ2ω ] with A[τ2ω ] 0 = A0 . In particular, if A0 is a maximal Hilbert algebra with an identity element e, then A[τ2ω ] is a GB∗ -algebra over U0 (A0 )1 e. Recall that U0 (A0 )1 denotes the unitization of the left von Neumann algebra U0 (A0 ). Notes Theorem 5.4.2 comes from Lemma 2.1 and Proposition 2.3 in [79]. For Lemma 5.4.3 see [81, Lemma 3.1 and Theorem 3.4]. Theorem 5.4.4 is due to Proposition 3.6 in [79] and Theorem 3.4 in [81]. Theorem 5.4.5 is due to Theorem 4.2 in [81], while Theorem 5.4.8 is due to Corollary 4.6 in [81]. Finally Theorem 5.4.10 is due to Theorems 3.2, 3.3 and 4.1 in [80].

Chapter 6

Generalized B*-Algebras: Unbounded *-Representation Theory

In representing a noncommutative C ∗ -algebra as a norm closed ∗-subalgebra of bounded linear operators on some Hilbert space, positive linear functionals play an important role. In ensuring that the involved ∗-representation is faithful, one requires that there are enough positive linear functionals in the sense that they separate the points of the C ∗ -algebra. That this is so for C ∗ -algebras is well known, and relies on the fact that the positive cone of a C ∗ -algebra is closed. It is not known if the positive cone is closed for GB∗ -algebras, in general, but we prove in Sect. 6.2 that the positive cone of a GB∗ -algebra A[τ ] is closed in some stronger topology T making A a GB∗ -algebra. In Sect. 4.2 (see Corollary 4.2.6), it was established (as in the case of C ∗ -algebras), that there are enough positive linear functionals on a commutative GB∗ -algebra, that separate its points. A partial analogue to this in the noncommutative case, we may say that it is given by Theorem 6.3.4. It is this result that we use to prove a noncommutative algebraic Gelfand–Naimark type theorem for GB∗ -algebras (see Theorem 6.3.5). More precisely, we show that any GB∗ algebra can be faithfully represented as an EC∗ -algebra (see Definition 5.3.1). For a topological analogue of the Gelfand–Naimark type theorem for GB∗ -algebras, see Theorem 6.3.11 and comments before Definition 6.3.8.

6.1 A Functional Calculus We have seen in Chaps. 3 and 4 that for the study of commutative GB∗ -algebras, an essential role is played by the Gelfand–Naimark type theorem (cf. Theorem 3.4.9) and the functional calculus given by Theorem 4.1.2. The Gelfand–Naimark type theorem just mentioned, can always be applied ‘locally’ in the noncommutative case too. Indeed, having a normal element x in an arbitrary GB∗ -algebra A[τ ], we may take a maximal commutative ∗-subalgebra B of A containing x. This

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Fragoulopoulou et al., Generalized B*-Algebras and Applications, Lecture Notes in Mathematics 2298, https://doi.org/10.1007/978-3-030-96433-7_6

127

128

6 Generalized B*-Algebras: Unbounded *-Representation Theory

is clearly closed and contains the identity e of A, so it is a commutative GB∗ algebra (see Proposition 3.3.19). In this way, an analogous functional calculus for noncommutative GB∗ -algebras is stated in Theorem 6.1.3 by just using, locally, Theorem 4.1.2. In this regard, see [48, p. 696, Section 4] together with Remarks 3.3.10 and 3.4.3. Dixon [48, Section 4], presented a noncommutative functional calculus for his ‘topological’ GB∗ -algebras (Definition 3.3.3), using locally the commutative functional calculus of Allan exhibited in Sect. 4.1. For this he used in the place of S ≡ σ (·), the spectrum σ D (·) (see (3.4.17)), taking into account Remarks 3.3.10 and 3.4.3.  As we have noticed in Chap. 3, throughout this book, we treat GB∗ algebras in the sense of Allan (Definition 3.3.2), unless mention is made to the contrary; the same we also do in the whole Chap. 6 and only where there is need, we apply Dixon’s strategy, based on the aforementioned Remarks 3.3.10 and 3.4.3. Recall that C∗ denotes the one point compactification of the complex plane C (see beginning of Sect. 2.3). Let A[τ ] be a GB∗ -algebra, x ∈ A and σ (x) the spectrum of x. As in Sect. 4.1, we put S ≡ σ (x) and denote by C1 (S) the set of all continuous C-valued functions f on the finite part of S ≡ σ (x), such that for a non-negative integer n, depending on f , the function λ →

f (λ) (1 + |λ|2 )n

extends to a bounded continuous function on the whole of σ (x). Lemma 6.1.1 Let A[τ ] be a GB∗ -algebra and x a normal element of A. Let C be a maximal commutative ∗-subalgebra of A containing x. Then, we have the following: (i) (ii) (iii) (iv)

C is a closed ∗-subalgebra of A[τ ]; C[τ C ] is a GB∗ -algebra having B0 ∩ C in B∗C as a maximal member. A[B0 ] ∩ C is a  · B0 -closed ∗-subalgebra of A[B0 ]; σA (y) = σC (y), for every y ∈ C.

Proof (i) is obvious. (ii) By (i) and Proposition 3.3.19, we conclude that C is a GB∗ -algebra, with B0 ∩C a maximal member in B∗C . (iii) It is easily seen that the set B0 ∩ C belongs to B∗A and that A[B0 ∩ C] = C ∩ A[B0 ]. The ∗-norm on A[B0 ∩ C] is just the restriction of the C ∗ - norm

6.2 Positive Elements

129

 · B0 of A[B0 ]. Thus, it follows that A[B0 ∩ C] is a Banach ∗-algebra from which (iii) follows. (iv) See Proposition 2.3.2.   Lemma 6.1.2 Let A[τ ] be a GB∗ -algebra and let C1 , C2 be maximal commutative ∗-subalgebras of A. If C ≡ C1 ∩ C2 , then C fulfills all the assertions (i)–(iv) of Lemma 6.1.1. Proof It follows easily from Proposition 3.3.19.

 

The following theorem gives a functional calculus for a not necessarily commutative GB∗ -algebra. For the notation applied, see Theorem 4.1.2. Theorem 6.1.3 (Dixon) Let A[τ ] be a GB∗ -algebra and x a normal element in A. Then, there exists a unique ∗-isomorphism  : C1 (S) → A, such that: (i) if u0 (λ) = λ , then u0 (x) = e; (ii) if u1 (λ) = λ, then u1 (x) = x; (iii) for every maximal commutative ∗-subalgebra C of A containing x and every f ∈ C1 (S), one has that

 f (x) ∈ C and f (x)(φ) = f  x (φ) , ∀ φ ∈ M0 , M0 being the carrier space of the commutative C ∗ -algebra C[B0 ∩ C]. Proof Fix a maximal commutative ∗-subalgebra B1 of A containing x. Then, Theorem 4.1.2 yields the existence of a unique ∗-isomorphism 1 : C1 (S) → B1 that fulfills (i) and (ii). Assume now that B2 is a second maximal commutative ∗-subalgebra of A containing x that gives rise to a corresponding ∗-isomorphism 2 : C1 (S) → B2 . Then, by Lemma 6.1.2, B3 := B1 ∩ B2 is a commutative GB∗ -subalgebra of A containing x. Thus, we obtain a third ∗-isomorphism 3 : C1 (S) → B3 . Applying the uniqueness claim of Theorem 4.1.2 for B1 , respectively B2 , we conclude that 1 = 3 , respectively 2 = 3 . Consequently,  = 1 satisfies (iii) for any maximal, commutative ∗-subalgebra B of A containing x.   Remark 6.1.4 In the sequel, we shall often apply (iii) of the preceding theorem, taking the Gelfand representation of a maximal commutative ∗-subalgebra of a GB∗ algebra A[τ ], containing a given normal element x of A.

6.2 Positive Elements Using the technique of Theorem 6.1.3(iii) and applying Theorem 3.4.9 and Corollary 3.4.10, one easily obtains the following Proposition 6.2.1 Let A[τ ] be a GB∗ -algebra, and x ∈ A. The following statements are equivalent:

130

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

6 Generalized B*-Algebras: Unbounded *-Representation Theory

x x x x

is normal and σ (x) ⊂ {λ ∈ C∗ : λ ≥ 0}; is normal and  x ≥ 0; = h2 , for some h ∈ H (A); = y ∗ y, for some y ∈ A.

Definition 6.2.2 If A[τ ] is a GB∗ -algebra, then we say that an element x ∈ A is positive, denoted by x ≥ 0, if x satisfies the equivalent conditions of Proposition 6.2.1. The set of positive elements is denoted by A+ . If x, y are in A, we write x ≤ y to mean that y − x ≥ 0 ⇔ y − x ∈ A+ . Note that, in the case when, A[τ ] is commutative, then Definition 6.2.2 coincides with Definition 4.1.3. Proposition 6.2.3 Let A[τ ] be a GB∗ -algebra and h a self-adjoint element in A. Then, there exist positive elements h+ , h− in A, such that h = h+ − h− and h+ h− = 0. Proof Take a maximal commutative *-subalgebra B of A containing h. Then, B is a commutative GB∗ -algebra with identity according to Propositions 2.3.2 and 3.3.19. Thus, applying Theorem 4.1.4(ii), there exist positive elements h+ , h− in B, such that the conclusion of our proposition is true.   Since the set of positive elements of a C ∗ -algebra forms a convex cone, it would be interesting to know if this is also the case for GB∗ -algebras, in general. That this is the case is proven below in Theorem 6.2.5. In order to prove it, we require the following lemma, which is an immediate consequence of Proposition 6.2.1(iv). Lemma 6.2.4 Let A[τ ] be a GB∗ -algebra. Then, for every x, y ∈ A+ , we have that xyx ∈ A+ . ∗ + Theorem 6.2.5 (Dixon)  Let +A[τ ] be a GB -algebra. Then, A is a convex cone, + (−A ) = {0}. with the property A

Proof If x ∈ A+ and λ > 0, then clearly λx ∈ A+ . Next we prove that x +y ∈ A+ , for every x, y ∈ A+ . We first show that (A[B0 ])+ = A[B0 ] ∩ A+ . Readily, (A[B0 ])+ ⊂ A[B0 ] ∩ A+ . Let now x ∈ A[B0 ] ∩ A+ . Then, its Gelfand transform  x is a bounded positive function on the maximal ideal space M0 of a maximal commutative ∗-subalgebra C of A containing x. Since the restriction of the Gelfand map to A[B0 ] ∩ C is the Gelfand representation of a maximal commutative ∗-subalgebra of the C ∗ -algebra A[B0 ], it follows that x ∈ (A[B0 ])+ , thereby establishing the equality (A[B0 ])+ = A[B0 ] ∩ A+ . If now h ∈ H (A), applying Remark 6.1.4, it follows from Theorem 3.3.9 that (e + h2 )−1 ∈ A+ ∩ A[B0 ]. Using this and Lemma 6.2.4, one has that, for every x, y ∈ A+ , z ≡ (e + y 2 )−1 x(e + y 2 )−1 ∈ A+ .

6.2 Positive Elements

131

Therefore, by appealing to Lemma 6.2.4 once more, we obtain that (e + z2 )−1 z(e + z2 )−1 ∈ A+ .

(6.2.1)

Applying again Remark 6.1.4, by Theorem 3.3.9 and Lemma 3.4.2, we have that (e + z2 )−1 z(e + z2 )−1 ∈ A[B0 ].

(6.2.2)

From (6.2.1) and (6.2.2), we obtain now that x1 ≡ (e + z2 )−1 (e + y 2 )−1 x(e + y 2 )−1 (e + z2 )−1 = (e + z2 )−1 z(e + z2 )−1 ∈ (A[B0 ])+ = A+ ∩ A[B0 ]. Since y ∈ A+ , again from (6.2.1) and (6.2.2), we conclude that (e + y 2 )−1 y(e + y 2 )−1 ∈ A+ ∩ A[B0 ] = (A[B0 ])+ . Consequently, as in the reasoning of (6.2.2), we have y1 ≡ (e + z2 )−1 (e + y 2 )−1 y(e + y 2 )−1 (e + z2 )−1 ∈ A[B0 ]. By Lemma 6.2.4, it follows now that y1 ∈ A+ ∩ A[B0 ] = (A[B0 ])+ . Since the positive cone of A[B0 ] is convex [45, p. 15, Proposition 1.6.1], we conclude that x1 + y1 ∈ (A[B0 ])+ ⊂ A+ , therefore by applying twice Lemma 6.2.4, we have x + y = (e + y 2 )(e + z2 )(x1 + y1 )(e + z2 )(e + y 2 ) ∈ A+ . Furthermore, it is clear by Proposition 6.2.1 that A+ a convex cone. This completes the proof.

 (−A+ ) = {0}. Thus, A+ is  

We next consider whether the positive cone A+ of a GB∗ -algebra A[τ ] is τ closed (see Theorem 6.2.11). Let A[τ ] be a GB∗ -algebra and let U be an absolutely convex 0-neighbourhood in A[τ ]. Fix an element y in A. Then, by the continuity of the map A[τ ] → A[τ ] : x → xy, we have that for every x in A there exists a 0-neighbourhood Vx , such that Vx y ⊆ U . Furthermore, using the 0-neighbourhood Vx and fixing now an element x in A, we obtain that for every y in A there is a 0-neighbourhood Wx,y , such that xWx,y ⊆ Vx , concluding thus, that xWx,y y ⊆ Vx y ⊆ U . Since B0 is a bounded subset of A, there exists δx,y ≡ δ(x, y) > 0, such that δx,y B0 ⊂ Wx,y . Let S denote the absolute convex hull of a subset S of A. Then, taking into account the preceding discussion

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6 Generalized B*-Algebras: Unbounded *-Representation Theory

we have that N(δ) := 



δx,y xB0 y ⊂ U.

(6.2.3)

x,y∈A

Now, we define a topology T on A to be the topology given by the base of neighbourhoods N(δ) , where δ ranges over all strictly positive functions of x, y in A. It is clear from (6.2.3) that the topology T is finer than the topology τ .  Observe that N(δ) is absorbent (since e ∈ B0 ) and absolutely convex from its construction; therefore T is a locally convex topology. Theorem 6.2.6 If A[τ ] is a GB∗ -algebra, then A[T ] is a barrelled and bornological GB∗ -algebra, with B0 (T ) = B0 (τ ), i.e., the greatest member B0 of B∗A , is the same with respect to both topologies T and τ on A. ∗ Proof If δx,y = δy ∗ ,x ∗ , for all x, y ∈ A, we have that N(δ ∗ ) = N(δ)∗ , which implies that the involution on A is T -continuous. Given z ∈ A and δ > 0, we define δz (x, y) := δzx,y , for all x, y ∈ A. Hence, (cf. (6.2.3)),

zN(δz ) = 



δzx,y zxB0 y

x,y∈A

⊂



δw,y wB0 y

w,y∈A

= N(δ). Therefore, the left multiplication is continuous. Similarly, the right multiplication is continuous. We now show that the conditions (i)–(iii) of Definition 3.3.3 are satisfied. 1 (i) Since B0 (τ ) ⊂ δe,e N(δ), we get that B0 (τ ) is T -bounded. Furthermore, since the topology τ is weaker than the topology T , it follows that B0 (τ ) is T -closed, and hence B0 (τ ) ∈ BA[T ] . This implies that B0 (τ ) ⊂ B0 (T ). If B ∈ BA[T ] , then B is T -bounded, and therefore, since the topology τ is weaker than the τ topology T , we obtain that B is also τ -bounded. Therefore, the τ -closure B τ of B is in BA[τ ] , so that B ⊂ B ⊂ B0 (τ ). It follows that B0 (τ ) is the greatest member of BA[T ] with respect to set inclusion, and hence B0 (T ) ⊂ B0 (τ ), implying that B0 (T ) = B0 (τ ). (ii) Since A[τ ] is a GB∗ -algebra and B0 (T ) = B0 (τ ), we obtain that the element (e + x ∗ x)−1 , x ∈ A, sits in A[B0 (τ )] = A[B0 (T )] and this implies symmetry of A[T ].

6.2 Positive Elements

133

(iii) Since A[τ ] is a GB∗ -algebra, A[B0 (τ )] is complete, and hence A[B0 (T )] (being equal to A[B0 (τ )]) is also complete. It follows that A[T ] is a GB∗ algebra. The fact that A[T ] is barrelled and bornological follows from the proof of Theorem 4.3.16 by applying similar steps.   Remark 6.2.7 In the sense of Definition 4.3.1 (see also Remark 4.3.2(2)) the topologies T and τ of Theorem 6.2.6 are “equivalent”. Clearly, this does not mean that the identity map idA : A[τ ] → A[T ] is a homeomorphism; it simply means that the C ∗ -algebras A[B0 (τ )] and A[B0 (T )] (through which A[τ ] and A[T ] are studied) are identical (see also comments after Corollary 6.3.7). Let E, F, G be topological vector spaces. A bilinear map f : E × F → G is called hypocontinuous, if f is separately continuous and for each bounded subset B of E and each 0-neighbourhood W in G, there is a 0-neighbourhood V in F , such that f (B × V ) ⊆ W ; see [131, p. 89]. In this regard, see also [74, p. 358]. Every locally convex algebra having jointly continuous multiplication, hence every Fréchet locally convex algebra, has hypocontinuous multiplication. Lemma 6.2.8 (Dixon) Let A[τ ] be a barrelled locally convex algebra. Then, given a bounded subset B and a 0-neighbourhood U of A[τ ], there exists a 0neighbourhood V in A[τ ], such that BV ⊆ U and V B ⊆ U. Proof We may choose the 0-neighbourhood U to be closed and absolutely convex. Consider the sets V1 ≡ {x ∈ A : B · x ⊆ U }, V2 ≡ {x ∈ A : x · B ⊆ U }. Note that, for every y ∈ A, the sets B · y, and y · B are bounded from the separate continuity of multiplication of A[τ ]. Therefore, B · y ⊆ λU for some λ > 0. This implies λ−1 y ∈ V1 , which shows that V1 is absorbing. Moreover, using the definition of V1 and the fact that U is absolutely convex, it is easily seen that V1 is absolutely convex too. Finally, using again the separate continuity of multiplication of A[τ ] and the closedness of U we conclude that V1 is also closed. Hence, V1 is a barrel. Analogously for V2 . Taking now V ≡ V1 ∩ V2 , we have V to be a barrel and so a 0-neighbourhood in A[τ ], such that BV ⊆ U and V B ⊆ U. Observe that V is nonempty. Indeed, given that B is bounded there is λ > 0, such that B ⊆ λU , hence λ1 e ∈ V1 ∩ V2 ≡ V , with e the identity in A. So the proof is complete.  

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6 Generalized B*-Algebras: Unbounded *-Representation Theory

An immediate consequence of Theorem 6.2.6 and Lemma 6.2.8 is the following Corollary 6.2.9 For every 0-neighbourhood U with respect to τ , there is a 0neighbourhood V with respect to τ , such that B0 V ⊂ U and V B0 ⊂ U.

([A])

We shall refer to the previous property as property [A] of the topology T . Property [A] is equivalent to the following: If (xλ )λ∈ , (yλ )λ∈ are nets in A, such that xλ ∈ B0 , ∀ λ ∈  and yλ → 0, then xλ yλ → 0 and yλ xλ → 0.  Observe that the property [A] is a weaker property than that of multiplication being hypocontinuous; that is every locally convex algebra having jointly continuous multiplication, hence every Fréchet locally convex algebra, has property [A]. The multiplication of every barrelled locally convex algebra is hypocontinuous [74, Theorem 2, p. 360]. Lemma 6.2.10 Let A[τ ] be a GB∗ -algebra, such that τ satisfies property [A]. If (yμ )μ∈M is a net in A+ converging to zero, then yμ 1/2 converges also to zero. Proof Let " f (λ) =

1 if 0 ≤ λ ≤ 1 λ−1/2 if λ > 1

" and

g(λ) =

λ1/2 − λ if 0 ≤ λ ≤ 1 0 if λ > 1.

We have that |f (λ)| ≤ 1 and |g(λ)| ≤ 1, for all 0 ≤ λ < ∞, and therefore f (yμ ), g(yμ ) ∈ B0 for all μ ∈ M (see Theorem 6.1.3 and Remark 6.1.4). Furthermore, λ1/2 = λf (λ) + g(λ), ∀ 0 ≤ λ < ∞; hence, again applying the last reference, we conclude that yμ 1/2 = yμ f (yμ ) + g(yμ ), ∀ μ ∈ M. Since the topology τ has the property [A], and since yμ → 0 and f (yμ ) ∈ B0 , for all μ ∈ M, it follows that yμ f (yμ ) → 0, i.e., for every 0-neighbourhood U , we have that yμ f (yμ ) ∈ U , for μ large enough. Therefore, yμ 1/2 ∈ U + B0 , for μ large enough. If V is a 0-neighbourhood, then there exists ε > 0 such that εB0 ⊂ 12 V .

6.2 Positive Elements

135

Let U = 12 ε−1 V . Since ε−1 yμ → 0, the above observation implies that ε−1 yμ 1/2 ∈ U + B0 =

1 −1 ε V + B0 , 2

for μ large enough. Therefore, yμ 1/2 ∈

1 1 1 V + εB0 ⊂ V + V = V , 2 2 2

for μ large enough, and so yμ 1/2 → 0.

 

Theorem 6.2.11 If A[τ ] is a GB∗ -algebra, such that τ satisfies property [A], then A+ is τ -closed. Proof Suppose that (xμ )μ∈M is a net in A+ and xμ → x with respect to the topology τ and x ∈ / A+ . Since the involution of A[τ ] is continuous, we obtain that x ∈ H (A). Hence, by Proposition 6.2.3, there are elements x + and x − in A+ , such that x = x + − x − and x + x − = 0. By separate continuity of multiplication, x − xμ x − → x − xx − = x − x + x − − (x − )3 = −(x − )3 ∈ −A+ . Let yμ = x − xμ x − + (x − )3 , for all μ ∈ M. By Lemma 6.2.4 and Theorem 6.2.5, we obtain that yμ ∈ A+ and yμ ≥ (x − )3 ∈ A+ , for all μ ∈ M. Furthermore, yμ → 0. Let y = (x − )3 . By the proof of Lemma 6.2.4, it follows that (e + y)−1 yμ (e + y)−1 ≥ (e + y)−1 y(e + y)−1 ≥ 0, for all μ ∈ M. By separate continuity of multiplication, it follows that (e + / A+ , it follows that x − = 0, therefore y > 0. y)−1 yμ (e + y)−1 → 0. Since x ∈ −1 −1 Hence, (e + y) y(e + y) = 0. We show that (e + y)−1 y(e + y)−1 ∈ B0 .

(6.2.4)

The elements y and (e + y)−1 are self-adjoint and commuting. Thus, we may consider a maximal commutative ∗-subalgebra C of A[τ ] containing them. Then, C[τC ] is a GB∗ -algebra with B0 ∩ C a maximal member in B∗C (Lemma 6.1.1(ii)). Using the commutative Gelfand–Naimark type Theorem 3.4.9 for C[τ C ], we have that x (ϕ) := ϕ (x)}, C = C = { x : C0 → C∗ : ϕ →  up to an algebraic ∗-isomorphism. We now show that ϕ (y) = ∞, where y as above.

For this we apply Lemma 3.4.4(ii), to C[τC ], with y in the place of x, (e+y 2)−1 =  (e + y ∗ y)−1 in the place of y and that ϕ ∈ M(C0 ), such that ϕ(e + y 2 )−1 = 0 (note that (e + y 2 )−1 ∈ C0 from Lemma 3.4.2 and that C0 is a C ∗ -algebra, since C is

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6 Generalized B*-Algebras: Unbounded *-Representation Theory

commutative).

But then from Lemma 3.4.4(ii) we obtain ϕ (y) = ∞. This shows that  y ∈ C M(C0 ) , therefore, y ∈ C0 . In the same way (taking in the place of x the element (e + y)−1 and in the place of y the element (e + y 2 )−1 ), we prove that

 ϕ ((e + y)−1 ) = ∞, hence (e + y)−1 ∈ C M(C0 ) , i.e., (e + y)−1 ∈ C0 . Finally (e + y)−1 y(e + y)−1 ∈ C0 = C[C ∩ B0 ] and since (e + y)−1 y(e + y)−1 is nonzero self-adjoint and B0 is balanced we conclude that (e + y)−1 y(e + y)−1 ∈ B0 and so (6.2.4) is proved. Let zμ = (e+y)−1 yμ (e+y)−1 , for all μ ∈ M. Let z ≡ (e+y)−1 y(e+y)−1 ∈ B0 . Then, zμ → 0, zμ ≥ 0 and zμ ≥ z ≥ 0, with 0 = z ∈ B0 . Put  := M × N, with N a directed set of positive integers. The set M becomes directed by defining (λ1 , n1 ) ≥ (λ2 , n2 ), if and only if, λ1 ≥ λ2 and n1 ≥ n2 . Then, a new net (wλ ) can be defined as follows

w(μ,n) = zμ +

1 e, with λ = (μ, n) ∈ . n

By the functional calculus (Remark 6.1.4) and Theorem 3.4.9, every wλ is invertible 1/2 −1/2 and (wλ−1 )1/2 = (wλ )−1 , which we write as wλ . We therefore have that −1/2

wλ ≥ z > 0, wλ

∈ A+ and wλ → 0.

1/2

By Lemma 6.2.10, it follows that wλ → 0. Using again the last reference and −1/2 −1/2 applying Lemma 6.2.4, one easily obtains that e ≥ wλ zwλ > 0, i.e., −1/2 −1/2 1/2 wλ zwλ ∈ B0 . Since wλ → 0 and the fact that the topology τ has property −1/2 −1/2 −1/2 1/2 1/2 [A], we obtain that wλ z = (wλ zwλ )wλ → 0. Now (e + wλ )−1 wλ ∈ B0 , for all λ ∈ , and therefore, by appealing again to the fact that the topology τ has property [A], it follows that

1/2  −1/2 (e + wλ )−1 z = (e + wλ )−1 wλ (wλ z) → 0. Moreover, (e + wλ )−1 ∈ B0 and z ∈ B0 , and so (e + wλ )−1 z ∈ B0 . Consequently, since the topology τ has property [A], wλ (e + wλ )−1 z → 0, implying that

 z = (e + wλ )−1 z + wλ (e + wλ )−1 z → 0. Hence, z = 0, which contradicts the fact that z = 0. Therefore x ∈ A+ and this completes the proof.   From the comment  after Corollary 6.2.9 and Theorem 6.2.11, we have the following Corollary 6.2.12 For every Fréchet GB∗ -algebra A[τ ], the convex cone A+ is τ closed.

6.3 *-Representations of GB*-Algebras

137

By Theorem 6.2.11 and the fact that topology T has property [A], we have the following result. Theorem 6.2.13 If A[τ ] is a GB∗ -algebra, then the cone A+ of positive elements in A[τ ] is T -closed. Notes Almost all results in this chapter have been obtained by P. G. Dixon in [48]. Not all locally convex GB∗ -algebras have property [A]. As an example, the ∗-algebra of all bounded linear operators on a Hilbert space is a GB∗ -algebra in the weak-operator topology, which does not satisfy property [A] and multiplication is only separately continuous. Hence, for every GB∗ -algebra A[τ ] we define a stronger topology T making A a (locally convex) GB∗ -algebra of Dixon (Definition 3.3.5), under which the positive cone A+ is T -closed.

6.3 *-Representations of GB*-Algebras The theory of ∗-representations of C ∗ -algebras is well understood. For instance, every C ∗ -algebra has a faithful ∗-representation onto a C∗ -algebra of bounded linear operators on some Hilbert space. In this section, we consider a generalization of this result for C∗ -algebras to the context of GB∗ -algebras. We recall the Gelfand–Naimark–Segal ∗-representation of a positive linear functional on a ∗-algebra. Let f be a positive linear functional on a ∗-algebra A. Then,   Nf = x ∈ A : f (x ∗ x) = 0 is a left ideal of A. If λf : A → A/Nf is the natural quotient map, the quotient space   λf (A) = λf (x) := x + Nf : x ∈ A is a pre-Hilbert space with inner product λf (x), λf (y) = f (y ∗ x), ∀ x, y ∈ A. We denote by Hf the Hilbert space obtained by the completion of the pre-Hilbert space λf (A). We define a ∗-representation πf0 of A by πf0 (x)λf (y) = λf (xy), ∀ x, y ∈ A and denote by πf the closure of πf0 . In this regard, we have the following Proposition 6.3.1 Let f be a positive linear functional on a ∗-algebra A. Then, there exists a closed ∗-representation πf of A in the Hilbert space Hf and a linear

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6 Generalized B*-Algebras: Unbounded *-Representation Theory

map λf of A on the domain D(πf ) of πf , such that λf (A) is dense in D(πf ) with respect to the graph topology tπf and πf (x)λf (y) := λf (xy), ∀ x, y ∈ A. The pair (πf , λf ) is uniquely determined by f up to unitary equivalence. We call the triple (πf , λf , Hf ) the GNS-construction for f . For more details, the reader is referred to [85, Theorem 1.9.1]. For a GNS like representation theorem in the context of ∗-bimodules, see [139], where an algebraic model for ∗-bimodules is presented and Hilbert space representations of ∗-bimodules are defined and studied. We define now the direct sum ⊕i∈I πi of a family of closed ∗-representations (πi )i∈I of a ∗-algebra.   Definition 6.3.2 Let πi : i ∈ I be a family of closed ∗-representations of a ∗algebra A. The algebraic direct sum of the ∗-representations πi ’s, denoted by π, is defined on the pre-Hilbert space Dπ =

 i∈I

 Dπi := (ξi )i∈I ∈ i∈I Dπi , with ξi = 0, for at most finitely many i ∈ I .

The inner product on Dπ and π(x) on Dπ , x ∈ A, are given as follows $

 % ξi , ηi i , ∀ (ξi )i∈I , (ηi )i∈I ∈ Dπ and (ξi )i∈I ), (ηi )i∈I ) := i∈I



 π(x) (ξi )i∈I := πi (x)ξi i∈I , ∀ x ∈ A and (ξi )i∈I ∈ Dπ . Considering now the Hilbert spaces Hi ’s, the completions of the pre-Hilbert spaces Dπi ’s, i ∈ I , we take the Hilbert space direct sum of Hi ’s, given by   ξi 2 < ∞ . Hπ = ξ = (ξi )i∈I ∈ i∈I Hi , i∈I

We clearly have that Dπ is dense in Hπ . Now, the closure  π (see comments before Example 5.1.3), of the algebraic direct sum ∗-representation π, is called the direct   sum of the closed ∗-representations πi : i ∈ I , and it is denoted by ⊕i∈I πi . πi , therefore π is closed, if and only if, all πi ’s are closed Note that  π = ⊕i∈I  (see [136, p. 213, 8.3.]). Thus, since each πi , i ∈ I , is closed, that is πi =  πi , we obtain that π =  π , consequently the algebraic direct sum ∗-representation π is closed too.

6.3 *-Representations of GB*-Algebras

139

Theorem 6.3.5 below is an extension of the classical noncommutative Gelfand– Naimark theorem to GB∗ -algebras in the context of O∗ -algebras. In order to prove this, we require the following two results.

 Lemma 6.3.3 Let π be a ∗-representation of a GB∗ -algebra A[τ ] and let U π(A) := {π(x) ∈ π(A) ∩ B(H) : π(x) ≤ 1} be the closed unit ball of π(A). Then π(B0 ) ⊆ U π(A) . Proof If x ∈ B0 and f is a positive linear functional on A, then f (x ∗ x) ≤ f (e), since A[B0 ] is a C ∗ -algebra. The map A # z → π(z)ξ, ξ ∈ C, ξ ∈ D(π), is also a positive linear functional on A, and therefore π(x)∗ π(x)ξ, ξ ≤ ξ, ξ . Hence, π(x) ≤ 1.   If A is a C ∗ -algebra, then f is continuous and f  = f (e) (cf. Proposition 4.2.3). The theorem that follows is well known for C ∗ -algebras and is concluded by a similar proof; for completeness’ sake we give its proof. Theorem 6.3.4 (Dixon) Let A[τ ] be a GB∗ -algebra. The following hold: (i) If x ∈ H (A) \ A+ , then there is a positive linear functional f on A, such that f (x) < 0; (ii) for every nonzero x ∈ A+ , there is a positive linear functional f on A, such that f (x) > 0. Proof (i) By Theorem 6.2.13, we obtain that A+ is a T -closed convex set. By the Hahn– Banach theorem, there is a linear functional f on A, such that f (x) < 0 and f (A+ ) ≥ 0. (ii) This is an immediate consequence of (i) and its proof.   As the case is with C ∗ -algebras, it will be illustrated in the proof of Theorem 6.3.5 that Theorem 6.3.4 is crucial in constructing faithful ∗-representations on GB∗ algebras. Besides, it is evident that Theorem 6.3.4 provides a GB∗ -algebra with enough positive linear functionals in order to separate its points; in this regard, see also Corollary 4.2.6. The next Theorem 6.3.5 offers an algebraic noncommutative Gelfand–Naimark type theorem in the context of GB∗ -algebras. Theorem 6.3.5 (Dixon) Every GB∗ -algebra A[τ ] has a faithful closed ∗representation π as an EC∗ -algebra on

D(π).  Furthermore, the ∗-representation π can be chosen, such that π(B0 ) = U π(A) . Proof We construct a faithful ∗-representation π in the same manner as one would in the C ∗ -algebra case. Let P(A) be the set of all positive linear functionals on A. For every f ∈ P(A), we define πf as in Proposition 6.3.1. Let π be the (algebraic) direct sum ∗-representation of the family {πf : f ∈ P(A)}. From Proposition 6.3.1, each πf , f ∈ P(A), is closed; so by the comments following Definition 6.3.2, we conclude that π is also closed. We show now that

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6 Generalized B*-Algebras: Unbounded *-Representation Theory

π is faithful. Let x = 0. This implies that x ∗ x = 0 too. Indeed, suppose that x ∗ x = 0. Then, from the Cauchy–Schwartz inequality we obtain f (x) = 0, for all f ∈ P(A). On the other hand, x = h + ik, with h, k ∈ H (A) and since every f ∈ P(A) preserves involution, we conclude that all f (h), f (k) are real numbers. So, f (x) = 0 implies f (h) = 0 = f (k), for all f ∈ P(A). Finally, Theorem 6.3.4 gives h = 0 = k, i.e., x = 0, a contradiction. Thus, having x ∗ x = 0, from Theorem 6.3.4(2), there will exist f ∈ P(A), with f (x ∗ x) = 0. Consequently, πf (x)λf (e)2 = f (x ∗ x) = 0. Letting now ξ = (ξg )g∈P (A) , where for g = f, ξf = λf (e) and for every other g ∈ P(A), ξg = 0, it follows that π(x)ξ = 0, therefore π(x) = 0. Hence, π is injective.



 We now show that π(B 0 ) = U π(A) . By Lemma 6.3.3, π(B0 ) ⊂ U π(A) , so it remains to prove that U π(A) ⊂ π(B0 ). We first show that, for any h ∈ H (A) with π(h) ≤ 1, we obtain hB0 ≤ π(h). Suppose that π(h) ≤ 1 and let f ∈ P(A). Let ξ = (ξg )g∈P (A) with  ξg :=

λf (e)

if g = f

0

if g = f

.

Then, f (x) = π(x)ξ, ξ , for all x ∈ A. Since π(h) ≤ 1, we obtain that π(h)ξ 2 ≤ ξ 2 ; therefore, f (h2 ) = π(h)ξ 2 ≤ ξ 2 = f (e), i.e., f (e − h2 ) ≥ 0, ∀ f ∈ P(A). By Theorem 6.3.4, it follows that e − h2 ≥ 0, i.e., h2 ≤ e. Consider the (commutative) GB∗ -subalgebra C[τ C ] of A[τ ] generated by the elements e, h and apply Theorem 3.4.9. Then, h2 ≤  e. This implies that  h is a bounded continuous function on the Gelfand space of C0 = C[C ∩ B0 ]. It follows that h ∈ C0 ⊂ A[B0 ], such that hB0 ≤ 1. Thus, if h ∈ H (A) with π(h) ≤ 1, then h ∈ A[B0 ] and hB0 ≤ 1. From this, it follows that if h ∈ H (A) and π(h) ∈ B(Hπ ) (for Hπ , see Definition 6.3.2), then h ∈ A[B

0 ] with  hB0 ≤ π(h). Let x ∈ A with π(x) ∈ U π(A) , and x not necessarily in H (A). Then, π(x) ∈ B(Hπ ) and there exist h, k ∈ H (A), such that x = h+ik. Therefore, π(h) and π(k) are in B(Hπ ), and so from the above, it follows that h, k ∈ A[B0 ], consequently x ∈ A[B0 ]. Also, hB0 ≤ π(h) and kB0 ≤ π(k); hence xB0 ≤ hB0 + kB0 ≤ π(h) + π(k) =

1 1 π(x ∗ ) + π(x) + π(x ∗ ) − π(x) ≤ 2π(x). 2 2

Recall that π(x) ≤ xB0 , therefore  · B0 and π(·) are equivalent norms on A[B0 ]. Since A[B0 ] is a C ∗ -algebra under  · B0 , it is complete. Hence, A[B0 ] ∗ is a C ∗ -algebra too, under the C ∗ -norm

π(·).  By standard C -algebra theory, ·B0 = π(·). Therefore, if π(x) ∈ U π(A) , we get that xB0 ≤ 1, i.e., x ∈ B0 .

6.3 *-Representations of GB*-Algebras

141



 Thus, π(x) ∈ π(B0 ), proving that U π(A) ⊂ π(B0 ). Since π(B0 ) = U π(A) , it follows that π(A[B0 ]) = π(A)b := π(A) ∩ B(Hπ ), which implies that π(A) is symmetric and π(A)b is a C∗ -algebra. From this we conclude that π(A) is an EC∗ -algebra (see Definition 5.3.1). This completes the proof.   Remark 6.3.6 Note that the ∗-representation π of the preceding theorem was constructed by using the positive linear functionals of A and without involvement of any topological structure of A. Consequently, the construction of π depends only on the algebraic structure of the GB∗ -algebra A[τ ] and the same is also true for the



 −1 U π(A) . set π Combining Remark 6.3.6 with the equivalence of two locally convex topologies given in Definition 4.3.1 (see also Remark 4.3.2(2)), we obtain an extension of Corollary 4.3.10 in the noncommutative case. More precisely, we have the following Corollary 6.3.7 (Dixon) Any two locally convex GB∗ -topologies τ1 , τ2 on a ∗algebra A are equivalent in the sense that the maximal elements of the collections B∗A[τ1 ] , B∗A[τ2 ] coincide; i.e., B0 (τ1 ) = B0 (τ2 ) and thus A[B0 (τ1 )] = A[B0 (τ2 )]. Proof From Theorem 6.3.5 and Remark 6.3.6 we conclude that



 B0 (τ1 ) = π −1 U π(A) = B0 (τ2 ). Therefore τ1 ∼ τ2 on A, according to the references in Chap. 4, mentioned just before.   The preceding corollary provides us with very good information. Namely, if a ∗-algebra A becomes a GB∗ -algebra (of Allan) under two topologies τ1 , τ2 , then the C ∗ -algebras A[B0 (τ1 )], A[B0 (τ2 )] respectively, coincide. As we have noticed these C ∗ -algebras are the key tools through which the structure of the GB∗ algebras A[τ1 ], A[τ2 ] respectively, is investigated. In this sense, we may say that any two locally convex topologies on a ∗-algebra A that make it a GB∗ -algebra are equivalent, having thus an ‘analogue’, in the context of GB∗ -algebras, of the situation that we meet in C ∗ -algebras. In fact, according to the comments we have just mentioned, we can make the equivalence of τ1 , τ2 , even clearer by saying that τ1 ∼ τ2 , if and only if, the identity map A[B0 (τ1 )] → A[B0 (τ2 )] is bicontinuous, which equivalently means that  · B0 (τ1 ) =  · B0 (τ2 ) . Coming back to the result of Theorem 6.3.5, we recall that for every C ∗ algebra A, there is a bicontinuous faithful ∗-representation π : A → B, where B is a C ∗ -algebra on some Hilbert space H. In this regard, Theorem 6.3.5 only provides us with the existence of a faithful ∗-representation of a GB∗ -algebra onto an EC∗ -algebra, and not necessarily a bicontinuous faithful ∗-representation onto an EC∗ -algebra with a suitable locally convex topology. The question is therefore if

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6 Generalized B*-Algebras: Unbounded *-Representation Theory

one can obtain such a representation theorem for GB∗ -algebras. For this, we require the following notion of an AO∗ -algebra, which is a non-normed generalization of a C ∗ -algebra, first defined by G. Lassner in [103, Definition 4.3]. In this way, Theorem 6.3.11 below. provides us with a necessary and sufficient condition, under which we may say that we obtain a noncommutative Gelfand–Naimark type theorem for GB∗ -algebras. Definition 6.3.8 An AO∗ -algebra is a complete locally convex ∗-algebra, which is topologically ∗-isomorphic to an O∗ -algebra in the uniform topology τu defined in (5.2.4). We present an example of an AO∗ -algebra. For this we need the result that follows; for the topologies involved, see (5.2.1) and (5.2.4). Theorem 6.3.9 (Lassner) Let π : A[τ ] → L† (D)[τw ] be a continuous ∗representation of a barrelled locally convex ∗-algebra A[τ ], when L† (D) carries the weak topology τw . Then, π : A[τ ] → L† (D)[τu ] is also continuous, when L† (D) carries the uniform topology τu . Proof Let V0 be a τu -neighbourhood of 0 ∈ π(A) (for the notation see the discussion after (5.2.4)). Then, there exists a tπ -bounded subset M of D and ε > 0, such that   π(x) ∈ π(A) : pM (π(x)) := sup |ξ, π(x)η | ≤ ε ⊆ V0 . ξ,η∈M

Let      UM,ε = x ∈ A : pM (π(x)) ≤ ε = x ∈ A : |ξ, π(x)η | ≤ ε . ξ,η∈M

It is clear that the function x → ξ, π(x)η is continuous, for all ξ, η ∈ D, by hypothesis. Hence, the set {x ∈ A : |ξ, π(x)η | ≤ ε} is absolutely convex and closed, for all ξ, η ∈ D. Consequently, UM,ε is absolutely convex and closed. Evidently, UM,ε is also balanced. We show that UM,ε is absorbing. Let x ∈ A and suppose that |α| ≥ 1ε pM (π(x)), α ∈ C. Then, pM (π(x)) ≤ |α|ε, hence α1 x ∈ UM,ε , i.e., x ∈ αUM,ε ; hence, UM,ε is absorbing. It follows that UM,ε is a barrel of A[τ ] and since A[τ ] is barrelled, we have that UM,ε is a neighbourhood of 0 ∈ A. Since also π(UM,ε ) ⊆ V0 , we obtain the conclusion.   Note that Example 6.3.10(1) is due to Lassner, while Example 6.3.10(2) is due to Schmüdgen.

6.3 *-Representations of GB*-Algebras

143

Examples 6.3.10 (1) We show that every barrelled pro- C ∗ -algebra A[τ ] (hence, every σ − C ∗ algebra; see Definition 3.1.2) is an AO∗ -algebra. Let {pν }ν∈ denote a family of C ∗ -seminorms on A defining the topology τ . If Nν := {x ∈ A : pν (x) = 0}, ν ∈ , it is easily verified that Nν is a closed ∗-ideal of A, for all ν ∈ . Let Aν ≡ A/Nν , ν ∈ . Put x + Nν ν := pν (x), ∀ x ∈ A, λ ∈ . Then, Aν [ · ν ] is a C∗ -algebra [60, p. 130, 10.(2)], for all ν ∈ . Therefore, Aν is topologically ∗-isomorphic to a C ∗ -subalgebra Mν of bounded linear operators on some Hilbert space Hν , with norm  · ν . Denote this topological ∗-isomorphism by φν . Then, x + Nν ν = φν (x + Nν )ν , for all x ∈ A and ν ∈ . Let qν : A → Aν , ν ∈ , be the corresponding quotient map. Then, ψν := φν ◦ qν is a ∗-homomorphism from A to Mν , with ψν (x)ν = φν (x + Nν )ν = x + Nν ν = pν (x),

(6.3.5)

for all x ∈ A, ν ∈ . Let H denote the Hilbert space direct sum of all Hν , and let D denote the algebraic direct sum of all Hν . Observe  that D is a dense subspace of H. For every x ∈ A and ξ ∈ H, let π(x)ξ = ν ψν (x)ξν , where  ξ = ν ξν ∈ D; in both sums, only a finite number of summands are nonzero. Clearly, π is a ∗-homomorphism of A into L† (D). We show that π is a topological ∗-isomorphism. For every ν ∈ , let Uν 1 denote the closed unit ball in Hν . Let η ∈ Uν and η 1 = (ηλ )λ ∈ D, where 1 1 1 ηλ = 0, for all λ = ν and ηλ = η if λ = ν. Then, η = λ ηλ = η1 ∈ D and π(x)η = π(x)

 λ

  ηλ1 = ψλ (x)ηλ1 = ψν (x)η, ∀ x ∈ A. λ

Observe now that the set Uν is a tL† (D) -bounded subset of D, for all ν ∈ . Indeed, for ξ ∈ Uν and x ∈ A, we have ξ π(x) = ξ +π(x)ξ  = ξ +ψν (x)ξ  ≤ ξ (1+ψν (x)) ≤ 1+ψν (x),

so Uν is tπ - bounded (see notation before Example 5.1.3), for each ν ∈ , which equivalently means that Uν is a tL† (D) -bounded set, for all ν ∈ ; since π(A) is a closed O∗ -algebra (see Definition 6.3.2) the latter follows from comments around (5.2.4). Thus, the seminorms {pUν }ν∈ , belong to the family

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6 Generalized B*-Algebras: Unbounded *-Representation Theory

of seminorms defining the topology τu . In particular, (see also (6.3.5)) pUν (π(x)) = sup |ξ, π(x)η | = sup |ξ, ψν (x)η | ξ,η∈Uν

ξ,η∈Uν

(6.3.6)

= ψν (x)ν = pν (x), for all x ∈ A and ν ∈ . Therefore, if π(x) = 0, then pν (x) = 0, for all ν ∈ , consequently x = 0. Furthermore, (6.3.6) shows, on the one hand that π −1 : π(A)[τu π(A) ] → A[τ ] is continuous and on the other hand, that π : A[τ ] → L† (D)[τw ] is also continuous. The latter, according to Theorem 6.3.9, implies that π : A[τ ] → L† (D)[τu ] is continuous. Thus, π is a topological ∗-isomorphism from A[τ ] on π(A)[τu ] (see notation after (5.2.4)). The result now follows from Definition 6.3.8. ∗ (2) Let {Aν }ν∈ be a directed family of barrelled pro-C  -algebras, each one of them having an identity element. Let A[τ ] := ν∈ Aν be their product endowed with the product topology τ and the usual algebraic operations and involution defined coordinatewise. Then, A[τ ] is a barrelled pro-C ∗ -algebra; see respectively [123, Proposition 4.2.5(i)] and [60, 7.6 Examples, (2)]. Hence, from the preceding Example 6.3.10(1), the product algebra A[τ ] is an AO∗ algebra. For further information on AO∗ -algebras, see [104, 105, 133]. The following result, connects GB∗ -algebras with AO∗ -algebras, where both are generalizations of C ∗ -algebras. Namely, we may say that Theorem 6.3.11 gives in the context of normal Fréchet GB∗ -algebras, a noncommutative Gelfand–Naimark type theorem. Theorem 6.3.11 (Schmüdgen) A Fréchet GB∗ -algebra A[τ ] is an AO∗ -algebra, if and only if, the (proper, convex) cone A+ of the positive elements in A[τ ] is normal. In what follows, we give a proof of Theorem 6.3.11, and for this, we require the two lemmas and proposition that follow. The given proof is inspired from the proof of a result of K. Schmüdgen (see [133, Theorem 5.1]), which reads as follows: a barrelled topological ∗-algebra A[τ ] with an identity element is an AO∗ -algebra, if and only if, the cone A+ of positive elements in A[τ ] is normal in the ordered topological vector space H (A)[τ ]. The concept of normality is given just before the proof of Theorem 6.3.11. Lemma 6.3.12 Let M be an absolutely convex subset of a pre-Hilbert space D. Then, for every T = T † ∈ L† (D), sup |T ξ, ξ | ≤ sup |T ξ, η | ≤ 2 sup |T ξ, ξ |.

ξ ∈M

ξ,η∈M

ξ ∈M

6.3 *-Representations of GB*-Algebras

145

Proof The first inequality is obvious, and so we give the proof of the second one. Let γ = supξ ∈M |T ξ, ξ | and ξ, η ∈ M. Choose r ∈ R, such that T (eir ξ ), η ∈ R. Let 2ξ1 = eir ξ + η and 2η1 = eir ξ − η. Since M is absolutely convex, we obtain that ξ1 , η1 ∈ M. Therefore, 4|T ξ, η | = 4|T (eir ξ ), η | = |2T η, eir ξ + 2T (eir ξ ), η | = 4|T ξ1 , ξ1 + T η1 , η1 | ≤ 4|T ξ1 , ξ1 | + 4|T η1 , η1 | ≤ 8γ . Therefore |T ξ, η | ≤ 2γ = 2 supξ ∈M |T ξ, ξ |.

 

Lemma 6.3.13 Let M be an absolutely convex subset of a pre-Hilbert space D. If T ∈ L† (D), then sup |T ξ, ξ | ≤ sup |T ξ, η | ≤ 4 sup |T ξ, ξ |.

ξ ∈M

ξ ∈M

ξ,η∈M

Proof Let T = T1 + iT2 , where Ti = Ti† ∈ L† (D) for i = 1, 2. By Lemma 6.3.12, sup |Ti ξ, ξ | ≤ sup |Ti ξ, η | ≤ 2 sup |Ti ξ, ξ |, i = 1, 2.

ξ ∈M

ξ ∈M

ξ,η∈M

Therefore, sup |T ξ, η | = sup |(T1 + iT2 )ξ, η | ξ,η∈M

ξ,η∈M

= sup |T1 ξ, η + iT2 ξ, η | ξ,η∈M

≤ 2 sup |T1 ξ, ξ | + 2 sup |T2 ξ, ξ | ξ ∈M

ξ ∈M

≤ 4 sup |T ξ, ξ |, ξ ∈M

since supξ ∈M |Ti ξ, ξ | ≤ supξ ∈M |T ξ, ξ |, for i = 1, 2. O∗ -algebra

 

Proposition 6.3.14 For an M on D, the uniform topology τu on M can also be defined by the family of seminorms

pM (T ) := sup |T ξ, ξ |, T ∈ M, ξ ∈M

where M is a tM -bounded subset of D (for tM , see after (5.1.1)).

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6 Generalized B*-Algebras: Unbounded *-Representation Theory

Proof If M1 denotes the absolute convex hull of M, then M1 is tM -bounded if M is tM -bounded. Observe that sup |T ξ, η | ≤ sup |T ξ, η |, T ∈ M. ξ,η∈M

(6.3.7)

ξ,η∈M1

By Lemma 6.3.13, sup |T ξ, ξ | ≤ sup |T ξ, η | ≤ 4 sup |T ξ, ξ |, T ∈ M.

ξ ∈M1

ξ,η∈M1

ξ ∈M1

(6.3.8)

(T − T ) → 0, for all t -bounded subsets Therefore, if (Tα ) is a net in M with pM α M M of D, we get from (6.3.8) that

sup |(Tα − T )ξ, η | → 0. ξ,η∈M1

By (6.3.7), sup |(Tα − T )ξ, η | → 0, ξ,η∈M

for all tM -bounded subsets M of D. Conversely, if sup |(Tα − T )ξ, η | → 0, ξ,η∈M

(T − T ) → 0, for all for all tM -bounded subsets M of D, then it is clear that pM α tM -bounded subsets M of D.  

Let E be a (real or complex) vector space. A subset C of E is called a cone, if it is invariant under addition and scalar multiplication by strictly positive scalars and C ∩ (−C) = {0} [136, p. 20]. Let E[τ ] be a real locally convex space, such that the topology τ is determined by a family = {p} of seminorms. A cone C in E is called normal, with respect to τ , if (see [131, p. 215, 3.1 (e)] and [133, p. 220, Section 4]) p(x) ≤ p(x + y), ∀ x, y ∈ C and ∀ p ∈ .

(6.3.9)

In [90, p. 86] a normal cone is called a self-allied cone. According to the second reference, just before (6.3.9), the seminorms p ∈ , having the previous property, are called monotone or normal seminorms for τ . If C is a cone in a topological vector space E, the dual cone C of C is given by the set   f ∈ E : Ref (x) ≥ 0, for x ∈ C ,

6.3 *-Representations of GB*-Algebras

147

where Ref (x) denotes the real part of f (x); namely, C is the polar of −C with respect to the dual pair (E, E ) (see [131, p. 218, comments before Lemma 1]). + Recall that if A[τ ] is a Fréchet GB∗ -algebra, the subset  A +of positive elements + (−A ) = 0) convex cone of A is a τ -closed proper (the latter meaning that A in A[τ ]. Hence, A+ fulfills all the properties of a cone as given above, so that in what follows we shall refer to A+ as a cone of A. We are ready now to exhibit the proof of Theorem 6.3.11. Proof First suppose that A[τ ] is an AO∗ -algebra. Then, A[τ ] is topologically ∗isomorphic to an O∗ -algebra M on a pre-Hilbert space D, where M is equipped with the uniform topology τu . By Proposition 6.3.14, the uniform topology τu can be defined by the family of seminorms

(T ) = sup |T ξ, ξ |, T ∈ M, pM ξ ∈M

where M is a tM -bounded subset of D. For all T , S ∈ M+ and for all tM -bounded subsets M of D,



pM (T + S) = sup (T ξ, ξ + Sξ, ξ ) ≥ sup T ξ, ξ = pM (T ). ξ ∈M

ξ ∈M

Concerning the inequality in the previous relation, observe that T , S are positive operators, so that T ξ, ξ ≥ 0 and Sξ, ξ ≥ 0, for every ξ ∈ M. Therefore, M+ is normal. We prove now that A+ is normal in the ordered real locally convex space H (A)[τ ]. Let φ be the topological ∗-isomorphism from A[τ ] to M[τu ]. Consider

◦ φ} on A, where M runs in the set of t -bounded the family of ∗-seminorms {pM M subsets of D. Then, it is readily seen that the topology τ is equivalent to the topology induced on A by the preceding family of ∗-seminorms. Thus, for any a, b ∈ A+ , we have





◦ φ)(a) = pM (φ(a)) ≤ pM (φ(a) + φ(b)) = (pM ◦ φ)(a + b), ∀ M, (pM

i.e., (6.3.9) is fulfilled and this shows that A+ is normal in H (A)[τ ]. Conversely, suppose that A+ is normal in H (A)[τ ]. Recall that since A[τ ] is a Fréchet GB∗ -algebra, each positive linear functional on A[τ ] is continuous (see Corollary 7.2.3, in Sect. 7.2). The *-representation, π (algebraic) direct sum of the family {πf : f ∈ P(A)} (Definition 6.3.2), is a ∗-homomorphism of A onto π(A). We prove that it is also injective. Let π(x) = 0, x ∈ A. Then, f (x ∗ x) = 0, for all f ∈ P(A), therefore (by the Cauchy–Schwarz inequality) f (x) = 0, for all f ∈ P(A), hence x = 0; if x = 0, from Theorem 6.3.4 (see also discussion after it) we would have f (x) = 0, for some f ∈ P(A), a contradiction. So π is an algebraic ∗-isomorphism onto π(A).

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6 Generalized B*-Algebras: Unbounded *-Representation Theory

Furthermore, we show that π : A[τ ] → π(A)[τu ] is a topological ∗isomorphism, thereby proving that A[τ ] is an AO∗ algebra. Let Fξ,η (x) = π(x)ξ, η , ∀ x ∈ A. For every ξ, η in the domain of π, we observe that Fξ,η is a finite sum of linear functionals Fξf ,ηf , where ξf and ηf are of the form πf (y)λf (e) and πf (z)λf (e) respectively, with y, z ∈ A and f a positive linear functional on A. Then, Fξf ,ηf (x) = πf (z∗ xy)λf (e), λf (e) = f (z∗ xy). Multiplication in A[τ ] is jointly continuous and since f is also continuous, it follows from the above that each Fξf ,ηf is continuous and consequently Fξ,η too. Therefore, π is weakly continuous. It follows now from Theorem 6.3.9 that π : A[τ ] → π(A)[τu ] is continuous. We are ready to prove that π −1 : π(A)[τu ] → A[τ ] is continuous. Since π(A)[τu ] can be identified with H (π(A))[τu ] ⊕ iH (π(A))[τu ] as locally convex spaces, it is sufficient to prove that π −1 : H (π(A))[τu ] → H (A)[τ ] is continuous. Employ the GNS-construction (πf , λf , Hf ), for each f ∈ P(A) ⊂ A (see discussion after Proposition 6.3.1). Then, we have that f (x) = πf (x)λf (e), λf (e) = π(x)ξ, ξ , ∀ x ∈ A,

(6.3.10)

where π = ⊕f ∈P (A) πf and ξ = (ξg )g∈P (A+ ) , with ξg = 0, for g = f and ξf = λf (e). Denote by M the subset of the domain Dπ of π, consisting of all ξ defined as before, for a given fixed f ∈ P(A), every time. Consider the set   F ≡ f ∈ P(A) : f (x) = π(x)ξ, ξ with x ∈ A and ξ ∈ M .

(6.3.11)

Then, F is an equicontinuous subset of A . Indeed, let V = {λ ∈ C : |λ| < ε}, ε > 0, be a 0-neighbourhood in C. Let   U = x ∈ A : sup pM (π(x)) < ε , ξ ∈M

where pM (π(x)) = |π(x)ξ, ξ |, x ∈ A, and M a tM -bounded subset of D. Now if W = {T ∈ π(A) : pM (T ) < ε} is a τu -neighbourhood of 0 in π(A), we clearly have that U = π −1 (W ) and since π is τ − τu continuous, U is a 0-neighbourhood in A[τ ]. Moreover, for each x ∈ U and for each f ∈ F , we obtain |f (x)| = |π(x)ξ, ξ | < ε, x ∈ A. Therefore, f (U ) ⊆ V , for all f ∈ F . To continue we have to justify that F is weakly bounded: A[τ ] being a Fréchet locally convex space is barrelled [74, p. 214, Corollary], therefore F as an equicontinuous subset of A is weakly bounded [74, p. 212, Corollary].

6.3 *-Representations of GB*-Algebras

149

By the weak boundedness of F , it follows now (see also (6.3.11)) that sup π(x)ξ 2 = sup π(x ∗ x)ξ, ξ = sup f (x ∗ x) < +∞,

ξ ∈M

ξ ∈M

f ∈F

for all x ∈ A. Hence, M is tπ -bounded. Now let F0 be an equicontinuous subset of A . From [131, p. 219, Corollary 1], the normality of A+ in H (A)[τ ] (see (6.3.9)) is equivalent to the existence of an equicontinuous subset F1 of (A+ ) (see comments before the proof of Theorem 6.3.11), such that F0 ⊆ F1 − F1 . So, we have (see (6.3.10) and Proposition 6.3.14)

pF1 (x) := sup |f (x)| = sup |π(x)ξ, ξ | = pM (π(x)), x ∈ A+ . f ∈F1

ξ ∈M

Consequently,

pF0 (x) ≤ 2pF1 (x) = 2pM (π(x)), ∀ x ∈ A+ .

(6.3.12)

Again the normality of A+ in A[τ ], from the preceding reference in [131], equivalently means that the locally convex topology τ on A is the topology of uniform convergence on the equicontinuous subsets of (A+ ) . It is now clear from (6.3.12) and Proposition 6.2.3 that π −1 is τu − τ continuous when restricted to H (A)[τ ]. This completes the proof.   Notice that in the complex locally convex space A[τ ], the cone A+ + iA+ is normal, if and only if, A+ is a normal cone in the real ordered locally convex space H (A)[τ H (A) ]. Corollary 6.3.15 Let A[τ ] be a Fréchet GB∗ -algebra. The following statements are equivalent: (i) (ii) (iii) (iv)

the given algebra A[τ ] is an AO∗ -algebra; the cone A+ is normal with respect to the topology τ ; the cone A+ is normal with respect to the weak topology σ (A, A ); for every continuous, self-adjoint, linear functional on A[τ ], there are two continuous, positive, linear functionals f1 , f2 on A[τ ], such that f = f1 −f2 .

Proof (i) ⇔ (ii) follows from Theorem 6.3.11. (ii) ⇒ (iii) Every element f ∈ A , can be written in the form f = g1 +ig2 , where g1 , g2 are continuous self-adjoint linear functionals on A[τ ]. Thus, we obtain σ ≡ σ (A, A ) H (A) = σ (H (A), H (A) ). Hence, since every normal cone in H (A) is also weakly normal by the second part of [131, p. 220, Corollary 3], the implication (ii) ⇒ (iii) is proved. (iii) ⇒ (ii) It is known that the bounded sets of H (A), with respect to the locally convex topologies τ H (A) and σ (H (A), H (A) ), compatible with the dual pair

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6 Generalized B*-Algebras: Unbounded *-Representation Theory

(H (A), (H (A) ), are the same (cf. [74, p. 209, Theorem 3 and p. 198, Definition 1]). Moreover, H (A)[τ H (A) ] is a real ordered metrizable locally convex space, so that the weakly normal cone A+ is also normal with respect to τ ; see [90, Theorem 3.2.13 and the discusiion after it]. Thus, (iii) ⇒ (ii). (iii) ⇒ (iv) From the first part of [131, p. 220, Corollary 3], we have that A = + (A ) − (A+ ) , where (A+ ) is the dual cone of A+ (see comments after (6.3.9)), i.e.,   (A+ ) = f ∈ A : Ref (x) ≥ 0, ∀ x ∈ A+ .

(6.3.13)

From the preceding expression of A , we conclude that each f ∈ A is of the form f = f1 − f2 , with fi ∈ (A+ ) , i = 1, 2. We shall show that fi ∈ P(A), i = 1, 2, i.e., taking, for instance i = 1, we must prove that f1 (x ∗ x) ≥ 0, for all x ∈ A. From Definition 6.2.2, all elements x ∗ x, x ∈ A, belong to A+ , hence (6.3.13) implies that Ref1 (x ∗ x) ≥ 0, where it is readily seen that f1 (x ∗ x) = f1 (x ∗ x), that is f1 (x ∗ x) is a real number. It follows now from (6.3.13) that f1 ∈ P(A). Similarly, f2 ∈ P(A) and this proves (iv). (iv) ⇒ (iii) Let f ∈ A . Then, f = f1 + if2 , with f1 , f2 self-adjoint elements in A . Thus, from (iv), we conclude that each f ∈ A is a linear combination of continuous, positive, linear functionals of A. Let fi (i = 1, · · · , n) be one of them. It is then easily seen that Refi (x) ≥ 0, ∀ x ∈ A+ ; i.e., fi ∈ (A+ ) , (see (6.3.13)). Finally we obtain that A = (A+ ) − (A+ ) , from which follows that the cone A+ is weakly normal (see the first part of [131, p. 220, Corollary 3]). (iv) ⇔ (i) From what we have shown before, (iv) ⇔ (iii) ⇔ (ii) ⇔ (i) and this completes the proof.    An immediate consequence of Corollary 6.3.15 is Corollary 6.3.16, just below, that provides an analogue of Theorem 4.2.5 in the case of noncommutative GB∗ -algebras. For the classical C ∗ -algebra counterpart of the latter, see [122, Theorem 3.2.5].

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151

Corollary 6.3.16 Let A[τ ] be a Fréchet GB∗ -algebra. The following are equivalent: (i) A[τ ] is an AO∗ -algebra; (ii) for every self-adjoint, continuous linear functional f on A[τ ] there are two continuous, positive, linear functionals f1 , f2 on A[τ ], such that f = f1 − f2 . Corollary 6.3.17 Let A[τ ] be a Fréchet GB∗ -algebra, such that its cone A+ is normal. Let B[τ B ] be a closed ∗-subalgebra of A[τ ] containing the identity e of A. Then, B[τB ] is an AO∗ -subalgebra of A[τ ]. Proof The closed ∗-subalgebra B[τ B ] of A[τ ] is clearly Fréchet, but also a GB∗ algebra from Proposition 3.3.19. Moreover, B + ⊂ A+ and since A+ is normal, the same is true for B + . Hence, Corollary 6.3.15 yields that B[τ B ] is an AO∗ subalgebra of the AO∗ -algebra (Theorem 6.3.11)) A[τ ].   Corollary 6.3.18 The Arens algebra Lω [0, 1] is an AO∗ -algebra.  p Proof From Examples 3.3.16(5) A[τ ] ≡ Lω [0, 1] := 1≤p 0 and some τ -continuous seminorm q, such that p((e + xα∗ xα )−1 − (e + xβ∗ xβ )−1 ) = p((e + xα∗ xα )−1 (xβ∗ xβ − xα∗ xα )(e + xβ∗ xβ )−1 ) ≤ q((e + xα∗ xα )−1 )q((e + xβ∗ xβ )−1 )q(xβ∗ xβ − xα∗ xα ) ≤ γ (e + xα∗ xα )−1 (e + xβ∗ xβ )−1 q(xβ∗ xβ − xα∗ xα ) ≤ γ q(xβ∗ xβ − xα∗ xα ),

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 ] and y ≡ limα (e + xα∗ xα )−1 Thus, (e + xα∗ xα )−1 is a Cauchy net in A[τ  ]. Since, exists in A[τ e = (e + xα∗ xα )(e + xα∗ xα )−1 = (e + xα∗ xα )−1 (e + xα∗ xα ),

∀ α,

 ] and y = (e + x ∗ x)−1 . Also, (e + x ∗ x)−1 ∈ it follows that (e + x ∗ x)−1 ∈ A[τ Bτ and in a similar way we have that x(e + x ∗ x)−1 and (e + x ∗ x)−1 x belong to Bτ .  τ ] and xBτ ≤ x, for each (ii) Since U (A) ⊆ Bτ , it follows that A ⊆ A[B ∗ x ∈ A. From the theory of C -algebras (see, for example, [144, Proposition I.5.3]), we have x ≤ xBτ , for every x ∈ A. Hence, x = xBτ , for each x ∈ A, which implies that U (A) = Bτ ∩ A and A is a closed ∗-subalgebra of  τ ]. A[B (iii) Take an arbitrary B ∈ B∗ containing U (A). Since B is τ -closed, it follows A  τ ] ⊆ A[B]  and xB ≤ xBτ , for each x ∈ that Bτ ⊆ B, therefore A[B  τ ]. Let x ∈ A[B].  A[B By (i) we have  −1 1  τ ], x e + x ∗x ∈ A[B n

∀ n ∈ N and

          1 ∗ −1 1 1 ∗ −1   ∗    lim x e + x x − x  = lim xx x e + x x  n→∞  n→∞ n    n n B B     1 ∗ −1  1   ∗ ≤ lim xx xB  e + x x  n→∞ n   n B  −1    1 1   ≤ lim xx ∗ xB  e + x ∗ x  n→∞ n   n

Bτ

≤ lim

n→∞

1 xx ∗ xB = 0. n

 τ ] is  · B -dense in A[B].  Hence, A[B This completes the proof.

 

For another aspect of Lemma 7.1.1, see Lemmas 3.4.2(i) and 7.3.2. Theorem 7.1.2 Let A[ · ] be a C ∗ -algebra with identity e and τ a locally convex topology on A, such that τ ≺ · and A[τ ] is a locally convex ∗-algebra with jointly  ] is a GB∗ -algebra over B0 = Bτ . continuous multiplication. Then, the algebra A[τ

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7 Applications I: Miscellanea

 ] is a complete locally convex ∗-algebra, it suffices to show that Proof Since A[τ  τ ], for every x ∈ A[τ  ]. Bτ is a greatest member in B∗ and that (e + x ∗ x)−1 ∈ A[B A The latter follows from Lemma 7.1.1(i).  τ ] is a • A crucial point for showing that Bτ is greatest in B∗, is to show that A[B A ∗ C -algebra. For this consider the C ∗ -algebra A of all  · -bounded nets (xλ )λ∈ in A[ · ], where the index set  consists of a 0-neighbourhood basis for τ , and the C ∗ -norm is given by (xλ )λ∈ ∞ := supλ∈ xλ , (xλ )λ∈ ∈ A. Take now,

Ac := (xλ )λ∈ ∈ A : (xλ )λ∈ is a τ − Cauchy net and  A0 := (xλ )λ∈ ∈ Ac : τ − limλ xλ = 0 . It is easily checked that Ac is a closed ∗-subalgebra of the C ∗ -algebra A[ · ∞ ], hence a C ∗ -algebra, and that A0 is a closed ideal (hence a ∗-ideal) in Ac . Therefore, the quotient Ac /A0 is a C ∗ -algebra. Now, an element a ∈ Bτ is of the form τ − limλ xλ , where (xλ )λ∈ is a net in A with xλ  ≤ 1, for all λ ∈ . In other words, a = τ − limλ xλ , with (xλ )λ∈ in Ac . Thus, the following correspondence  τ ] : (xλ )λ∈ → τ − lim xλ ,  : Ac → A[B λ

is a well-defined continuous ∗-homomorphism, such that ker  = A0 and ((xλ )λ ) ≤ supλ xλ  = (xλ )λ ∞ , ∀ (xλ )λ ∈ Ac .  τ ].  from Ac /A0 onto A[B Observe that  induces an isometric ∗-isomorphism  −1  Indeed, since  is a ∗-homomorphism from a Banach ∗-algebra onto a C ∗ algebra, we have that 

 (xλ )λ + A0 , ∀ (xλ )λ + A0 ∈ Ac /A0 , (xλ )λ + A0 q ≤  where  · q is the quotient C ∗ -norm on Ac /A0 . On the other hand, from the very definitions we have 

 (xλ )λ + A0  ≤ (xλ )λ ∞ ≤ (xλ )λ + a0 ∞ + a0 ∞ ,  with a0 ∈ A0 . Hence, 

 (xλ )λ + A0  ≤ (xλ )λ + A0 q , ∀ (xλ )λ ∈ Ac .  This completes the proof of our claim.  τ ] is a C ∗ -algebra. Consequently, the Banach ∗-algebra A[B • We show now that Bτ is the greatest member in B∗. A

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161

Take an arbitrary B ∈ B∗ and h∗ = h ∈ B. Let C be a maximal commutative ∗A  ] containing h. Then, C is complete with respect to the topology subalgebra of A[τ induced by τ . We denote by B∗C the collection of all closed, bounded, absolutely convex subsets B1 of C satisfying: e ∈ B1 , B1∗ = B1 and B12 ⊆ B1 . Then, B∗C = {B2 ∩ C : B2 ∈ B∗}. We show that B ∩ C ⊆ Bτ ∩ C. Since C is commutative A and complete, it follows from Theorem 2.2.10, that B∗C is directed, so there exists B1 ∈ B∗C , such that (B ∩ C) ∪ (Bτ ∩ C) ⊆ B1 . So, since Bτ ∩ C ⊆ B1 , we obtain  τ ] ∩ C. xB1 ≤ xBτ ∩C = xBτ , ∀ x ∈ A[B  τ ] ∩ C is contained in the On the other hand, since the C ∗ -algebra C[Bτ ∩ C] = A[B Banach ∗-algebra C[B1 ], it follows from [144, Proposition I.5.3] that  τ ] ∩ C. xBτ = xBτ ∩C ≤ xB1 , ∀ x ∈ A[B Thus, we have  τ ] ∩ C. xB1 = xBτ , ∀ x ∈ A[B

(7.1.1)

Next we show that Bτ ∩ C = B1 . Take an arbitrary x ∈ C[B1 ] and n ∈ N. By Lemma 7.1.1(i), we obtain −1  −1  1  τ ], so that x e + 1 x ∗ x  τ ] ∩ C. ∈ A[B ∈ A[B x e + x ∗x n n The estimate     −1  −1      x e + 1 x ∗ x   = 1 xx ∗ x e + 1 x ∗ x − x     n n n B1 B1     1 1 ∗ −1   x ≤ xx ∗ xB1  x e +   n n B1      −1 1 ∗ 1  = xx ∗ xB1   e + nx x  n Bτ ≤

1 xx ∗ xB1 n

 τ ] ∩ C is  · B1 -dense in C[B1 ]. Therefore, by (7.1.1) we implies now that A[B conclude that  τ ] ∩ C = C[Bτ ∩ C] = C[B1 ], A[B and thus Bτ ∩ C = B1 .

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7 Applications I: Miscellanea

This implies that h ∈ B ∩C ⊆ Bτ ∩C. Consequently, we have shown that h ∈ Bτ , for each h = h∗ ∈ B. So,   τ ] and x2 = x ∗ xBτ ≤ 1, ∀ x ∈ B. A[B] ⊆ A[B Bτ Hence, B ⊆ Bτ and Bτ is a greatest member in B∗A . This completes the proof.

 

 ] The C ∗ -normed algebra A[ · ] that determines the locally convex ∗-algebra A[τ ∗ ∗ is not unique. For this reason, we denote by C (A, τ ) the set of all C -normed algebras A [ ·  ], such that  ], τ ≺  ·  and x = x, ∀ x ∈ A. A ⊆ A ⊆ A[τ Then C ∗ (A, τ ) is an ordered set under the following order: A1 [ · 1 ] ≺ A2 [ · 2 ], if and only if, A1 ⊆ A2 and x1 = x2 , ∀ x ∈ A1 .  τ ] is the largest element in C ∗ (A, τ ). It also implies Theorem 7.1.2 shows that A[B the following Corollary 7.1.3 Let A[ · ] be a C ∗ -algebra with identity e and τ a locally convex ∗-algebra topology on A, such that τ ≺  · . Then, the following statements are equivalent:    ] is a GB∗ -algebra over the closed unit ball U (A) := x ∈ A : x ≤ 1 (i) A[τ of the C ∗ -algebra A[ · ]. (ii) U (A) is τ -closed. Let now D be a dense subspace in a Hilbert space H, such that L† (D) is closed. Let τu be the uniform topology on L† (D) (see discussion after (5.2.4)). Suppose that M is a C ∗ -algebra on H, such that MD ⊆ D and let the multiplication on M be jointly continuous, with respect to the topology τu . Furthermore, suppose that  u ] is closed. If this does not happen, we simply consider the the O∗ -algebra M[τ  u ] in L† (D)[τu ]. Then, by Lemma 5.3.3 and Corollary 7.1.3 we τu -closure of M[τ obtain the following  u ] is a GB∗ -algebra over U (M). Corollary 7.1.4 M[τ Further, we investigate the representation theory of the complete locally convex  ] as in Corollary 7.1.3. Let A[τ  ]+ denote the τ -closure of the positive ∗-algebra A[τ + ∗  ]+ is a wedge, in the sense that, cone A in the C -algebra A[ · ]. Then, A[τ  ]+ , ∀ x, y ∈ A[τ  ]+ and ∀ λ ≥ 0. x + y and λx ∈ A[τ

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163

 ]+ is identical to the τ -closure of the algebraic wedge Moreover, A[τ n 

  ], k = 1, 2, . . . , n, n ∈ N .  ] := P A[τ xk∗ xk : xk ∈ A[τ k=1

 ] is said to be strongly positive if f (x) ≥ 0, for all A linear functional f on A[τ +  x ∈ A[τ ] . Furthermore, note that L† (D) is a locally convex ∗-algebra equipped with the weak topology τw (resp. strong∗ topology τs ∗ ) defined by the family pξ,η (·) :  ξ, η ∈ D of seminorms with pξ,η (T ) := |T ξ, η |, T ∈ L† (D) (resp. the family  ∗  pξ (·) : ξ ∈ D of seminorms with pξ∗ (T ) := T ξ  + T ∗ ξ , T ∈ L† (D)). We shall say that a ∗-representation π of a locally convex ∗-algebra A[τ ] is (τ − τw )-continuous (reps. (τ − τs ∗ )-continuous) if it is continuous from A[τ ] to π(A)[τw ] (resp. to π(A)[τs ∗ ]). In this regard, we have (for Pták function, see Definition 3.2.8) Lemma 7.1.5 The following statements are equivalent:  

 ]+  ]+ = {0}. (i) A[τ − A[τ

   τ ]+ = {0}.  τ ]+ − A[B (ii) A[B  τ ] is a C ∗ -norm. (iii) The Pták function pA[Bτ ] on the Banach ∗-algebra A[B  ]. (iv) There exists a faithful ∗-representation of A[τ  ]. (v) There exists a faithful (τ − τs ∗ )-continuous ∗-representation of A[τ Proof (i) ⇒ (v) Let F be the set of all τ -continuous strongly positive linear  ]. Let (πf , λf , Hf ) be the GNS-construction for f ∈ F . We functionals on A[τ put D(π) :=



 & λf (xf ) ∈ Hf : λf (xf ) = 0, ∀ f ∈ F, except for a finite number , f ∈F





  ], λf (xf ) ∈ D(π). π(a) λf (xf ) := λf (axf ) , ∀ a ∈ A[τ  ]. Then, it is easily shown that π is a (τ − τs ∗ )-continuous ∗-representation of A[τ ∗  We show that π is faithful. Take 0 = a ∈ H (A[τ ]) (i.e., a = a). Assume that  ]+ . Since A[τ  ]+ ∩ (−A[τ  ]+ ) = {0}, we have A[τ  ]+ ∩ {−a} = ∅. Then, a ∈ A[τ it follows from [35, Chap. II, §5, Proposition 4] that there exists a τ -continuous  ], such that f (a) > 0. Now assume that strongly positive linear functional f on A[τ + +   a ∈ A[τ ] . Since A[τ ] ∩ {a} = ∅, we can show in a similar way that there exists  ], such that f (a) < a τ -continuous strongly positive linear functional f on A[τ 0. Since πf (a)λf (e)|λf (e) = f (a) = 0, this implies that πf (a) = 0, and so  ] we have π(a) = 0 by considering π(a) = 0. Similarly, for any 0 = a ∈ A[τ  a = a1 + ia2 (a1 , a2 ∈ H (A[τ ])).

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7 Applications I: Miscellanea

(v) ⇒ (iv) This is trivial.  ]. By Lemma 7.1.1(i), (iv) ⇒ (iii) Let π be a faithful ∗-representation of A[τ  A[Bτ ] is a symmetric Banach ∗-algebra, hence hermitian (see discussion after Definition 3.1.2). Therefore, the Pták function pA[Bτ ] (x) := rA[Bτ ] (x ∗ x)1/2 , x ∈  τ ], is a C ∗ -seminorm (cf. [52, (33.1) Theorem]). In particular (Raikov criterion A[B for symmetry [127, (4.7.21) Theorem]), pA[Bτ ] (x) =

sup

[Bτ ]) ρ∈Rep(A

ρ(x), x ∈ A[Bτ ],

  τ ] denotes the set of all ∗-representations of A[B  τ ]. Suppose where Rep A[B

  pA[Bτ ] (x) = 0. Since π A[Bτ ] ∈ Rep A[Bτ ] , we have π(x) = 0, and so x = 0. Thus, pA[Bτ ] is a C ∗ -norm. (iii) ⇒ (ii) We first show that    τ ]+ . spA[Bτ ] (x) ⊆ R+ ≡ λ ∈ R : λ ≥ 0 , ∀ x ∈ A[B

(7.1.2)

 τ ]+ and a net (xα ) in A+ that converges to x, In fact, take an arbitrary x ∈ A[B  with respect to τ . Since A[Bτ ] is hermitian we have that spA[Bτ ] (x) ⊆ R. Let λ < 0. Notice that λ(λe − xα )−1 ∈ U (A), for every α. Then, for any τ -continuous  ] seminorm p on A[τ 

p λ(λe − xα )−1 − λ(λe − xβ )−1

 = |λ|p (λe − xα )−1 (xα − xβ )(λe − xβ )−1

 

≤ |λ|q (λe − xα )−1 q(xα − xβ )q (λe − xβ )−1 1 γ λ(λe − xα )−1 λ(λe − xβ )−1 q(xα − xβ ) |λ| γ q(xα − xβ ), ≤ |λ|

≤

 ]. It follows that for some constant γ > 0 and a τ -continuous seminorm q on A[τ −1 λ(λe − xα ) converges to an element y of Bτ with respect to τ , which implies that λ(λe − x)−1 exists and equals y. Hence, λ ∈ spA[Bτ ] (x). Thus, we have 

 τ ]+ ∩ − A[B  τ ]+ . Then, from (7.1.2), spA[Bτ ] (x) ⊆ R+ . Take an arbitrary x ∈ A[B it follows that spA[Bτ ] (x) = {0}, therefore pA[Bτ ] (x) = rA[Bτ ] (x) = 0. Since pA[Bτ ] is a norm, we have x = 0. 

 ]+ . Then, from Lemma 7.1.1(i)  ]+ ∩ − A[τ (ii) ⇒ (i) Take an arbitrary a ∈ A[τ

  τ ]+ ∩ − A[B  τ ]+ = {0}, which implies a = 0. it follows that a(e + a 2 )−1 ∈ A[B This completes the proof.  

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165

 ]. We define Let now π be a ∗-representation of A[τ   ]π := x ∈ A[τ  ] : π(x) ∈ B(Hπ ) D(rπ ) = A[τ b rπ (x) := π(x), ∀ x ∈ D(rπ ), where Hπ is a Hilbert space. Then, rπ is an unbounded C ∗ -seminorm (see  ], induced discussion before Definition 3.5.1) on the ∗-subalgebra D(rπ ) of A[τ by π.  ] and B any element of Lemma 7.1.6 Let π be a faithful ∗-representation of A[τ ∗ B  containing U (A). Then, the following statements hold: A

 τ ] ⊆ A[B]   ]π and π(x) ≤ xB , for all x ∈ (1) A ⊆ A[B ⊆ D(rπ ) = A[τ b  A[B]. In particular, π(x) = x = x, for all x ∈ A. B τ





   ] , and it is also uniformly dense in π A[τ  ]π . (2) π A[B] is τs∗ -dense in π A[τ b (3) Suppose that π is (τ − τw )-continuous. Then, 



 

  ]+ ⊆ L† D(π) + := T ∈ L† D(π) : T ≥ 0 . π A[τ Proof Statement (1) is easily checked.  ]. Then, it follows that (2) Take an arbitrary a ∈ A[τ √ −1 √ √ 1

 τ ], ∀ ε > 0 (e + εa ∗ a)−1 a = √ e + ( εa)∗ ( εa) ( εa) ∈ A[B ε and for each ξ ∈ D(π) 

  π (e + εa ∗ a)−1 a ξ − π(a)ξ 



  = επ (e + εa ∗ a)−1 π(a ∗ a 2 )ξ  

  ≤ επ (e + εa ∗ a)−1 π(a ∗ a 2 )ξ     = ε(e + εa ∗ a)−1 Bτ π(a ∗ a 2 )ξ    ≤ επ(a ∗ a 2 )ξ  −−→ 0, ε↓0





 ] . Take an arbitrary a ∈ A[τ  ]π . Then, since  τ ] is τs ∗ -dense in π A[τ so that π A[B b     

 π (e + εa ∗ a)−1 a ξ − π(a)ξ  ≤ επ(a ∗ a 2 )ξ , ∀ ξ ∈ D(π),

 it follows that limε↓0 π (e + εa ∗ a)−1 a = π(a) uniformly, which implies that 



 ]π . Since A[B   τ ] ⊆ A[B],  τ ] is uniformly dense in π A[τ we conclude (2). π A[B b

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7 Applications I: Miscellanea

 (3) This follows from (τ − τw )-continuity of π and the fact that π A+ ⊆

+ L† D(π) . This completes the proof.   Theorem 7.1.7 The following statements are equivalent:  ] is a GB∗ -algebra. (i) A[τ  ], such (ii) There exists a faithful (τ − τs ∗ )-continuous ∗-representation π of A[τ that τ ≺ rπ .  ] is a GB∗ -algebra over B0 . Since Proof (i) ⇒ (ii) Suppose A[τ  



 0 ]+ ∩ − A[B  τ ]+ ∩ − A[B  τ ]+ ⊆ A[B  0 ]+ = {0}, A[B Lemma 7.1.5 implies the existence of a faithful (τ −τs ∗ )-continuous ∗-representation  ]. Furthermore, since π(A[B  0 ]) is a C ∗ -algebra, Lemma 7.1.6(2) yields that of A[τ     



 0 ] = π A[τ  ]π and rπ (x) = π(x) = x  , π A[B b B 0

∀ x ∈ D(rπ ),

which implies τ ≺ rπ . (ii) ⇒ (i) Since τ ≺ rπ and π is (τ − τs ∗ )-continuous, it follows that τ and rπ are compatible, so one obtains that the completion Arπ of D(rπ )[rπ ] is embedded  ]. We denote by B0 the τ -closure of the closed unit ball U (Arπ ) of the C ∗ in A[τ algebra Arπ . Then, B0 ∈ B∗ and from Lemma 7.1.6(1) we have A

  ]π ⊆ B0 , ∀ B ∈ B∗, B ⊆ U A[τ b A

  ]π , is a greatest member in B∗ . Thus, from which implies that B0 = U A[τ b

A π   ] and this  ] is a GB∗ -algebra over U A[τ Theorem 7.1.2, we conclude that A[τ b completes the proof.    ] is a Fréchet ∗-algebra, the following are equivalent: Corollary 7.1.8 If A[τ  ] is a GB∗ -algebra. (i) A[τ  ] such that τ ≺ rπ . (ii) There exists a faithful ∗-representation π of A[τ  ] is (τ − Proof Every ∗-representation of the Fréchet locally convex ∗-algebra A[τ ∗ τ s )-continuous. Indeed, take an arbitrary ξ ∈ D(π) and put fξ (x) := π(x)ξ, ξ , x ∈ A. Then, fξ is a positive linear functional on the Fréchet ∗-algebra A[τ ], which is continuous by [51, Theorem 4.3] (see also [60, Theorem 15.5]). Furthermore, since the multiplication of a Fréchet ∗-algebra is jointly continuous, it follows that π is (τ − τs ∗ )-continuous. Hence, the assumption follows from Theorem 7.1.7.  

7.2 Continuity of Positive Linear Functionals of GB*-Algebras

167

Notes Almost for all the results of this section, the reader is referred to [12] and [62]. In particular, Lemmas 7.1.1, 7.1.5, 7.1.6, Theorem 7.1.7 and Corollary 7.1.8 can be found in [12, Section 2], while Theorem 7.1.2 comes from [62, Theorem 2.1] (see also [12, Theorem 2.2]). Related studies can be found in [22–25]. For the investigation of Case 2 (where the multiplication is considered separately continuous, see beginning of this section), we refer the reader to [12–14, 62].

7.2 Continuity of Positive Linear Functionals of GB*-Algebras In this section the presented results concern GB∗ -algebras of Dixon (Definition 3.3.5). It is well known that all positive linear functionals of a C ∗ -algebra are continuous and so the question is if one can extend this result to GB∗ -algebras. We shall see in Corollary 7.2.2 below that at least some GB∗ -algebras have this property. We also give examples of GB∗ -algebras, which admit at least one discontinuous positive linear functional, thereby showing that the conditions in Theorem 7.2.1 cannot be dropped. Theorem 7.2.1 (Dixon) If A[τ ] is a sequentially complete GB∗ -algebra satisfying property [A] of Section 6.2, then every positive linear functional on A is bounded. Proof Let f be a positive linear functional of A. We first show that if f is bounded on all bounded sequences of A, then f is bounded. So suppose that f is bounded on all bounded sequences of A. If f is not bounded, then there exists a bounded subset C of A, such that for every n ∈ N, there exists yn ∈ C, such that |f (yn )| > n. This implies that f is not bounded on the (yn ), which contradicts our claim. Therefore, it suffices to prove that f is bounded on all bounded sequences of A. Let (xn ) be a bounded sequence of positive elements of A. Also, let (λn ) be a sequence in l 1 with λn ≥ 0, for all n ∈ N. For every absolutely convex 0neighbourhood U in A, there exists k > 0, such xn ∈ kU , for all n ∈ N. that q Since (λn ) ∈ l 1 , there exists N ∈ N, such that i=p λi < 1k for all p, q ≥ N. Therefore, q 

λi xi ∈ U, ∀ p, q ≥ N.

i=p

 Since A is sequentially complete, it follows that ∞ i=1 λi xi converges to x ∈ A, say. Since A satisfies property [A], it follows from Theorem 6.2.11 that A+ is closed, hence x ∈ A+ . Consequently, ∞  i=n+1

λi xi = x −

n  i=1

λi xi ≥ 0, ∀ n ∈ N.

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7 Applications I: Miscellanea

Thus, n 

λi f (xi ) = f

i=1

n  i=1

∞     λi xi = f (x) − f λi xi ≤ f (x). i=n+1

Therefore, by the monotone convergence theorem for sequences of real numbers, ∞ 1 λ f (x n ) is convergent for all (λn ) in l . Hence, (f (xn )), n ∈ N, is a bounded n=1 n sequence. Now let (xn ) be a bounded sequence of self-adjoint elements of A. By taking the Gelfand representation of xn , we obtain that

 

− e + 2xn2 (e + xn2 )−1/2 ≤ xn ≤ e + 2xn2(e + xn2 )−1/2 ; hence, 

|f (xn )| ≤ f (e) + 2f xn2 (e + xn2 )−1/2 .

(7.2.3)

Using again the Gelfand representation of xn , we obtain that xn (e + xn2 )−1/2 ∈ B0 . Since A satisfies property [A], then for any 0-neighbourhood U in A, there exists a 0-neighbourhood V , such that B0 V ⊂ U . Due to the fact that (xn ) is a bounded sequence, there exists m ∈ N, such that xn ∈ mV for all n ∈ N. Therefore,

−1/2

−1/2 = xn e + xn2 xn ∈ B0 mV ⊂ mU, xn2 e + xn2 

for all m ∈ N, implying that xn2 (e + xn2 )−1/2 is a bounded sequence of positive

 elements in A. From the above, it follows now that f xn2 (e + xn2 )−1/2 is bounded. Hence, by (7.2.3), f (xn ), n ∈ N, is a bounded sequence. Finally, let (xn ) be any bounded sequence in A. Then, yn :=

1 1 (xn + xn∗ ) and zn := (xn − xn∗ ), 2 2i

define bounded sequences in A by continuity of involution. Since f (xn ) = f (yn ) + if (zn ), for all n ∈ N, it follows that the sequence f (xn ), n ∈ N, is bounded and this completes the proof.   Corollary 7.2.2 Every positive linear functional on a sequentially complete, bornological GB∗ - algebra is continuous. Proof Let A[τ ] be a GB∗ -algebra and f a positive linear functional on A. A sequentially complete bornological space is barrelled, and therefore A has hypocontinuous multiplication, therefore it has property [A]. By Theorem 7.2.1, it follows that f is bounded, and since A is bornological, f is continuous.   Since every Fréchet locally convex algebra is sequentially complete and bornological, we obtain the following result.

7.2 Continuity of Positive Linear Functionals of GB*-Algebras

169

Corollary 7.2.3 Every positive linear functional on a Fréchet GB∗ -algebra is continuous. In 1959, D-S. Sya proved that the above corollary holds for any Fréchet locally convex ∗-algebra with an identity element, i.e., he proved that every positive linear functional of a Fréchet locally convex ∗-algebra with identity is continuous [143] (see also [51, Theorem 4.3] and/or [60, Theorem 15.5]). Later, in 1978, G. Dales, with instruction from Dixon, proved that every positive linear functional of a Fréchet topological algebra, with identity and continuous involution, is continuous [42, Theorem 11.1]. In the next two examples, we will show that neither assumptions of sequential completeness nor the algebra being bornological can be dropped from Corollary 7.2.2, thereby giving examples of GB∗ -algebras admitting at least one discontinuous positive linear functional. Example 7.2.4 ([48, Example 8.4]) Let A = L∞ [0, 1] be equipped with the topology defined by the family of seminorms being the Lp -norms  · p , p ∈ N. Since Lω [0, 1] is a GB∗ -algebra over the C ∗ -algebra L∞ [0, 1] (see Example 3.3.16(5)), then {B ∩ L∞ [0, 1] : B ∈ BLω [0,1] } has a largest element B0 with respect to set inclusion, which is the closed unit ball of L∞ [0, 1]. Furthermore, the bounded part of L∞ [0, 1] is a C ∗ -algebra, namely itself, and symmetric. Therefore, A = L∞ [0, 1] is a GB∗ -algebra of Dixon (i.e., a locally convex GB∗ -algebra, in the sense of Definition 3.3.5). We define a positive linear functional F on A by 

1

F (f ) = 0

f (t) dt, ∀ f ∈ A. t (log(t) − 1)2

Let N ∈ N. We define fq,N (t) = min{(4t)−1/2q , N}, ∀ q = 1, 2, 3, · · · . It easily follows that  fq,N q ≤

1

(4t)−1/2 dt

0

1/q

 dt =

 lim

s→0+ s

1

(4t)−1/2 dt

1/q dt = 1.

Now, 

η

F (fq,N ) ≥ 0

N N , dt = 2q log(N) + log(4) + 1 t (log(t) − 1)2

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7 Applications I: Miscellanea

where η = 4N12q . The last expression goes to infinity as N → ∞, for every q. If F is continuous, then there exists p ∈ N and a constant K > 0, such that |F (f )| ≤ Kf p for all f ∈ A. This gives a contradiction for f = fp,N , upon letting N → ∞. The topology defined above is metrizable, hence bornological and therefore, in light of Corollary 7.2.2, A cannot be sequentially complete. Example 7.2.5 ([48, Example 8.3]) Suppose that X is a countably compact, noncompact, completely regular space, being either locally compact or first countable. Let A ≡ Cc (X) be the commutative pro-C ∗ -algebra of all continuous C-valued functions on X endowed with the compact-open topology ‘c’ [112, p. 52, Proposition 12.2, a)]. As such, A is a GB∗ -algebra (see Examples 3.3.16(1)), that also has property [A] according to the observation in  after Corollary 6.2.9. Therefore, Corollary 7.2.2 yields that every positive linear functional on A is continuous. But, each character of A is the point evaluation corresponding to an element x ∈ βX, ˇ the Stone–Cech compactification of X. On the other hand, from a known result of Gelfand–Kolmogorov (cf. [68, Theorem 1]), one concludes that the character of A corresponding to an element x ∈ βX\X is discontinuous, and since every character of A is ∗-preserving (see [60, Corollary 9.3(1)] (Michael)) is clearly a positive linear functional. So, A accepts discontinuous positive linear functionals. Since A is complete, and therefore sequentially complete, this is attributed to the fact that A cannot be bornological. The above example also illustrates that ∗-representations of GB∗ -algebras are not continuous, in general. Recall, in this regard, that all ∗-representations of a C ∗ algebra are continuous. Moreover, notice that R.M. Brooks [37, p. 17, Example 6.1] has given an example of a discontinuous ∗-representation on a pro-C ∗ -algebra, hence a GB∗ -algebra (see also [60, Sect. 17, pp. 196–197]). Notes All the results of this Section are due to P.G. Dixon and the reader can find them in [48, 50].

7.3 Every GB*-Algebra has a Bounded Approximate Identity It is well known that every C ∗ -algebra and every pro-C ∗ -algebra (Inoue) [60, p. 137, Theorem 11.5] has a bounded approximate identity. Below, we extend this result to GB∗ -algebras. If A[τ ] is a topological ∗-algebra, we recall that a bounded approximate identity of A is a bounded net (eα ) in A, such that lim eα x = x = lim xeα , ∀ x ∈ A. α

α

7.3 Every GB*-Algebra has a Bounded Approximate Identity

171

For the definition of hypocontinuity, used in the statement of Theorem 7.3.1, see comments right after Remark 6.2.7. The following Theorem 7.3.1 is valid for every GB∗ -algebra in the sense of Dixon (Definition 3.3.3). Theorem 7.3.1 (Bhatt) Let A[τ ] be a GB∗ -algebra. The following hold: (i) if the multiplication of A[τ ] is hypocontinuous and I is a closed two-sided ideal of A[τ ], then every bounded approximate identity of I ∩ A[B0 ] is also an approximate identity of I ; (ii) every GB∗ -algebra A[τ ] admits a bounded approximate identity. Proof (i) Since A[τ ] has hypocontinuous multiplication and I is a two-sided ideal of A, it follows that I also has hypocontinuous multiplication with respect to the relative topology inherited from the topology τ on A. Let (eα ) be a bounded approximate identity of I ∩ A[B0 ]. By the continuity of addition and hypocontinuity of multiplication on I , it follows that for any given 0-neighbourhood U in I , we can choose 0-neighbourhoods V and V1 in I , such that V1 B0 + V + V ⊂ U . By Proposition 3.3.19, I is a GB∗ -algebra over the C ∗ -algebra I ∩ A[B0 ]. Let x ∈ I , and let xn = x(e + n1 x ∗ x)−1 , for all n ∈ N. From Theorem 6.3.5 A[τ ] is represented as a unital ∗-algebra of closed operators sitting in π(A) and having a common domain, dense in some Hilbert space, Hπ . Besides, A[B0 ] coincides with the ∗-algebra of bounded operators of π(A). Thus, the element x ∈ A can be identified with a closed densely defined operator T from π(A). By Rudin [129, Theorem 13.13(a)], the operator T (I + T ∗ T )−1 is bounded, therefore the elements xn , n ∈ N, corresponding to the bounded operators T (I + n1 T ∗ T )−1 , n ∈ N, belong to A[B0 ]. Hence, finally we have that xn ∈ I ∩A[B0 ], for all n ∈ N. Moreover, xn → x, as follows from a similar argument τ

as in the proof of Lemma 7.1.1(iii); consequently there exists n0 ∈ N, such that x −xn ∈ V1 , for all n ≥ n0 . Similarly, there exists n1 ∈ N, such that xn −x ∈ V , for all n ≥ n1 . Let m = max{n0 , n1 }. Since xn ∈ I , for all n ∈ N, and (eα ) is a bounded approximate identity of I ∩ A[B0 ], it follows that xn eα → xn , for all n ∈ N. Therefore, by taking n ≥ m, there exists an α0 , such that for all α ≥ α0 , we have xeα − x = (x − xn )eα + (xn eα − xn ) + (xn − x) ∈ V1 B0 + V + V ⊂ U. Hence, xeα → x, for all x ∈ I . Similarly, eα x → x, so we conclude that (eα ) is a bounded approximate identity for I .

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(ii) Let A[τ ] be a GB∗ -algebra. We recall that A can be equipped with the topology T , as defined just before Theorem 6.2.6. By the latter theorem, A[T ] is a barrelled GB∗ -algebra, and therefore A[T ] has hypocontinuous multiplication (see [131, p. 89, 5.2] and/or [74, p. 360 Theorem 2]). Furthermore, Theorem 6.2.6 also informs us that B0 (T ) = B0 (τ ). Since A[B0 ] is a C ∗ -algebra, it admits a bounded approximate identity (eα ). By applying (i) to the case where I = A, it follows that (eα ) is also a bounded approximate identity for A[T ]. Since the topology T on A is stronger than the given topology τ , we get that (eα ) is a bounded approximate identity of A[τ ] too.   Note that from the third paragraph of the proof of Theorem 7.3.1(i), thanks to Theorem 6.3.5 and to [129, Theorem 13.13(a)], we conclude the following result, which extends in the noncommutative case Lemma 3.4.2(i); recall that in the latter reference A0 = A[B0 ], since A[τ ] is commutative (cf. Lemma 3.3.7(ii)). In this regard, the reader may also look at Lemma 7.1.1, for another aspect of the content of the lemma that follows. Lemma 7.3.2 Let A[τ ] be a GB∗ -algebra. Then, for every x ∈ A, one has that the element x(e + x ∗ x)−1 belongs to the C ∗ -algebra A[B0 ]. Notes Theorem 7.3.1 is due to S.J. Bhatt, and can be found in [18].

7.4 Inverse Limits and Quotients of GB*-Algebras  In this section, unless stated otherwise, a GB∗ -algebra will mean a GB∗ algebra in the sense of Definition 3.3.3 (Dixon). A GB∗ -algebra A[τ ], in the sense of Definition 3.3.3 (Dixon), has the property that the corresponding family of sets BA , defined just before Definition 3.3.3, has a largest element B0 with respect to set inclusion, which gives rise to a C ∗ -algebra A[B0 ] contained in A (see also Remarks 3.3.4 and 3.3.10). In this section, we show that we can avoid having to make use of the set B0 in that we can characterize GB∗ algebras by using more algebraic techniques (see Theorem 7.4.5). This will lead to some information of inverse limits of GB∗ -algebras. In order to prove Theorem 7.4.5 below, we require Proposition 7.4.1 and Lemma 7.4.3 that we first prove. Proposition 7.4.1 Let A[ · ] be a normed ∗-algebra with identity element e and let B[ ·  ] be a C ∗ -algebra, which is also a ∗-subalgebra of A with e ∈ B. If

7.4 Inverse Limits and Quotients of GB*-Algebras

173

(e + x ∗ x)−1 ∈ B, for all x ∈ A, then A = B as algebras (and not necessarily as normed algebras).  · ] denote the completion of A[ · ]. Taking into consideration Proof Let A[  it follows immediately from [127, Theorem 4.8.5] that sp (x) = that e ∈ B ⊆ A, A spB (x) for all x ∈ B. It follows from this and the hypothesis that



 0∈ / spA (e + x ∗ x)−1 = spB (e + x ∗ x)−1 −1

for all x ∈ A. Therefore e + x ∗ x = (e + x ∗ x)−1 ∈ B, hence x ∗ x ∈ B for all x ∈ A. Therefore, if x ∈ A, then, since x=

 1 (x + e)∗ (x + e) − (x − e)∗ (x − e) + i(x + ie)∗ (x + ie) − i(x − ie)∗ (x − ie) , 4

it follows that x ∈ B, i.e., A = B.

 

Corollary 7.4.2 Let A[ · ] be a normed ∗-algebra, which is also a GB∗ -algebra, such that the topology on A[B0 ] defined by  ·  is weaker than the  · B0 -topology. Then, there exists a norm  ·  equivalent to  · , such that A[ ·  ] is a C ∗ -algebra. Proof Since A[ · ] is a GB∗ -algebra, we have that (e + x ∗ x)−1 ∈ A[B0 ], for all x ∈ A, and that A[B0 ] is a C ∗ -algebra with respect to the Minkowski functional  · B0 on B0 (see Theorem 3.3.9). Take,  ·  :=  · B0 . By Proposition 7.4.1, it follows that A = A[B0 ]. Hence, by [144, Proposition I.5.3], xB0 ≤ x, for all x ∈ A. Since A[ · ] is a GB∗ -algebra, it follows that the  · -topology is weaker than the  · B0 -topology on A[B0 ] = A. Therefore, the norms  ·  and  ·  are equivalent on A.   The following lemma is Corollary 4.8.4 of [127]. Lemma 7.4.3 If A[ · ] is a C ∗ -algebra and  ·  is another norm on A with respect to which A is a ∗-normed algebra, then x2 ≤ x ∗  x , for all x ∈ A. We let   [−e, e] = h ∈ H (A) : −e ≤ h ≤ e , and E the absolute convex hull of [−e, e]. When the natural ordering is Archimedean, then the vector space E generated by E becomes a normed ∗-space when equipped with the Minkowski functional  ·  ≡  · E of E. For the proof of the next theorem, we still need the following Lemma 7.4.4 Let A[τ ] be a GB∗ -algebra. Then, the C ∗ -algebra A[B0 ] is the linear hull of [−e, e]. Proof Let h ∈ A[B0 ] be self-adjoint. Take the commutative C ∗ -subalgebra of A[B0 ], say C, generated by {h, e}. Then, C ∼ = C(X), where X is the maximal ideal space of C. We can therefore identify h with  h ∈ C(X). Since  h is bounded, there

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7 Applications I: Miscellanea

exists k > 0, such that | h(ϕ)| ≤ k, for each character ϕ ∈ X. By the functional calculus in C ∗ -algebras, it follows that −ke ≤ h ≤ ke. Since now every x ∈ A[B0 ] is a linear combination of self-adjoint elements, we conclude that A[B0 ] is the linear hull of [−e, e].   The following theorem is the promised characterization of GB∗ -algebras using more algebraic techniques. Theorem 7.4.5 Let A[τ ] be a topological ∗-algebra with identity e. The following conditions are equivalent: (i) A[τ ] is a GB∗ -algebra with underlying C ∗ -algebra A[B0 ]; (ii) A[τ ] has a ∗-subalgebra B, which is a C ∗ -algebra with respect to some norm  · , the element (e + x ∗ x)−1 ∈ B, for all x ∈ A, and the closed unit ball U (B) of B is τ -bounded; (iii) E is, with respect to some norm  ·  , a C ∗ -algebra, such that E is τ -bounded, and (e + x ∗ x)−1 ∈ E, for all x ∈ A. If (i), (ii) and (iii) hold, then A[B0 ] = E = B. Proof (i) ⇒ (iii) We shall show that E = A[B0 ]. By Lemma 7.4.4, A[B0 ] is the linear hull of [−e, e], therefore is contained in the linear hull of E (the latter being the absolutely convex hull of [−e, e]), which yields that A[B0 ] ⊆ E. We now show that E ⊆ A[B0 ]. Let h ∈ [−e, e]. Consider the maximal commutative GB∗ -subalgebra C of A[τ ] containing h. By Remark 3.4.3(3), C is algebraically ∗-isomorphic to the ∗-algebra C of C∗ -valued functions on the compact Hausdorff space X corresponding to the Gelfand space of the commutative C ∗ -algebra C0 = C[B0 ∩ C] with an identity element (see Lemma 6.1.1, which will be used again right after). Under this isomorphism, h is identified with an element  such that −1 ≤   h ∈ C, h(ϕ) ≤ 1, for every character ϕ ∈ X. By Remark 3.3.10(a), h ∈ C0 = C[B0 ∩ C] ⊂ C. It follows that hB0 =  h∞ ≤ 1, which yields h ∈ B0 . This proves E ⊆ A[B0 ] and so E = A[B0 ]. Therefore, (e + x ∗ x)−1 ∈ A[B0 ] = E, for all x ∈ A (see Theorem 3.3.9). Lastly, since E = A[B0 ] and B0 is τ -bounded, it follows that E is τ -bounded. (iii) ⇒ (ii) This follows by taking B = E. (ii) ⇒ (i) It remains to show that U (B) is the largest element of BA , with respect to set inclusion. Let B ∈ BA , h ∈ B ∩ H (A), and C (as above) the maximal commutative ∗subalgebra of A containing h (with h now in B ∩ H (A)). Then, C ≡ C ∩ B is a commutative C ∗ -subalgebra of B[ · ], while C is a τ -closed ∗-subalgebra of A[τ ]. Take the closed unit ball U (C) of C[ · ] and if H ≡ {e, hn : n ∈ N}, denote by Y the absolutely convex hull of H · U (C) in C.

7.4 Inverse Limits and Quotients of GB*-Algebras

175

We prove that Y is τ -bounded. Let Ln denote the mapping x → hn x of C[ · ] into A[τ ], and let L = {Ln : n ∈ N}. It follows from the continuity of multiplication and the τ -boundedness of H ⊆ B, that L is pointwise bounded, i.e., {Ln (x) : x ∈ C, n ∈ N} is τ -bounded. Since C[ · ] is of second category, it follows from a generalization of the uniform boundedness theorem [131, Theorem, p. 83] that L is uniformly bounded. In particular, for every 0-neighbourhood V in A[τ ], there exists λ > 0, such that H · U (C) ⊆ λV . So, H · U (C) is τ -bounded and since Y is the absolutely convex hull of H · U (C), it follows easily that Y is τ -bounded. Furthermore, simple calculations show that Y 2 ⊆ Y , Y ∗ = Y and e ∈ Y . If Y denotes the vector space generated by Y , then we show that Y is a normed ∗-algebra, when equipped with the Minkowski functional  · Y of Y . We already know that  · Y is a ∗-seminorm on Y, therefore it remains to show that ·Y is a norm. Let U denote a τ -neighbourhood base of 0 ∈ A and assume that xY = 0. Since Y is τ -bounded, for every U ∈ U, there exists a scalar μ(U ) > 0, such that Y ⊆ μ(U )U . Therefore, x ∈ λY ⊆ (λ · μ(U ))U , for all λ > 0 and for all U ∈ U. Since τ is Hausdorff, it follows that x = 0. Observe that U (C) ⊆ Y ⊆ Y, hence C ⊆ Y. Moreover, by our assumption (ii), (e + y ∗ y)−1 ∈ B, for all y ∈ Y. On the other hand, by the construction of Y, it is evident that Y ⊆ C and C as a C ∗ -algebra is symmetric, so that (e+y ∗ y)−1 ∈ C, for each y ∈ C. Thus finally, (e +y ∗ y)−1 ∈ B ∩C = C, for all y ∈ Y. Proposition 7.4.1 implies now that Y = C. Therefore, by Lemma 7.4.3, x2 ≤ x ∗ Y xY , for all x ∈ Y . Consequently, since Y ∗ = Y , we obtain that x ∗ Y ≤ 1, for all x ∈ Y . Hence, x2 ≤ 1, for all x ∈ Y . Therefore, Y ⊆ U (C) and so h ∈ Y ⊆ U (C) ⊆ U (B), i.e., B ∩ H (A) ⊆ U (B). If x ∈ B, then by the above, x ∗ x ∈ B ∩ H (A) ⊆ U (B). Since B[ · ] is a 1 ∗ C -algebra, it follows that x = x ∗ x 2 ≤ 1, i.e., x ∈ U (B). This proves that U (B) is the largest element of BA , with respect to set inclusion.   From Theorem 7.4.5, we obtain the following results. Corollary 7.4.6 Let Aα [τα ], α ∈ J , be a family of GB∗ -algebras, where J is a directed index set. Then, the product A = α∈J Aα , with respect to the product topology and algebraic operations and involution defined coordinatewise, is a GB∗ algebra. Furthermore,  A[B0 ] = (xα )α∈J ∈ A : xα ∈ Aα [(B0 )α ] and sup xα α < ∞ . α∈J

Proof Since (B0 )α is τα -bounded for all α ∈ J , it follows easily that B0 = α∈J ((B0 )α ) is bounded in A with respect to the product topology. On the other hand, it is clear that  B = (xα )α∈J ∈ A : xα ∈ Aα [(B0 )α ] and sup xα α < ∞ α∈J

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is a C ∗ -algebra with respect to the norm x = sup xα α , x = (xα )α ∈ A. α

Let eα be the identity element of Aα , for all α ∈ J , and let x = (xα )α ∈ A. Clearly, e = (eα )α is the identity element of A. Since Aα [τα ] is a GB∗ -algebra over the C ∗ -algebra Aα [(B0 )α ], for all α ∈ J , it follows that (eα + xα∗ xα )−1 ∈ (B0 )α ⊂ Aα [(B0 )α ], for all α ∈ J . Consequently, (e + x ∗ x)−1 ∈ B for all x ∈ A. Therefore, by Theorem 7.4.5, A[τ ] is a GB∗ -algebra and A[B0 ] = B.   By similar reasoning as in the proof of Corollary 7.4.6, one obtains the following Corollary 7.4.7 (Kunze) Let Aα [τα ], α ∈ J , denote a family of GB∗ -algebras, for which J is a directed index set. Let hαβ : Aβ [τβ ] → Aα [τα ] denote a family of continuous ∗-homomorphisms with hαβ (eβ ) = eα , for α ≤ β, where eα is the identity element of Aα , for each α ∈ J , such that (Aα [τα ], hαβ )(α≤β in J ) is an inverse system of GB∗ -algebras. Then, the inverse limit

 A[τ ] = lim hαβ Aβ [τβ ] ←

is a GB∗ -algebra with  A[B0 ] = (xα )α∈J ∈ A : xα ∈ Aα [(B0 )α ] and sup xα α < ∞ . α∈J

Every pro-C ∗-algebra A[τ ] with identity is an inverse limit of C ∗ -algebras. Therefore, by Corollary 7.4.7, we retrieve the result that every pro-C ∗ -algebra is a GB∗ -algebra, as shown in Example 3.3.16(1). Recall that the quotient algebra of a closed ideal in a C ∗ -algebra is also a C ∗ algebra. The following result shows that this is true for all GB∗ -algebras too. Theorem 7.4.8 (Kunze) Let A[τ ] be a GB∗ -algebra and I a closed two-sided ideal in A[τ ]. Then, I is a ∗-ideal and the quotient algebra A/I , equipped with the quotient topology τq , is a GB∗ -algebra with underlying C ∗ -algebra A[B0 ]/(I ∩ A[B0 ]). Proof Observe that, since A[B0 ] is norm complete and I is τ -closed in A, I0 ≡ I ∩ A[B0 ] is a norm closed two-sided ideal of the C ∗ -algebra A[B0 ]. Therefore, I0 is a ∗-ideal of A[B0 ], and A[B0 ]/I0 is a C ∗ -algebra. Let x ∈ I . Then, by Remark 3.3.10(b), we obtain that x(e + x ∗ x)−1 ∈ I . On the other hand, arguing as in the proof of Theorem 7.3.1(i), we have that x(e + x ∗ x)−1 ∈ A[B0 ] and finally x(e + x ∗ x)−1 ∈ I0 . Then, ∗

(e + x ∗ x)−1 x ∗ = x(e + x ∗ x)−1 ∈ I0 .

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177

Therefore, x ∗ = (e + x ∗ x)(e + x ∗ x)−1 x ∗ ∈ I . It follows that I is a ∗-ideal of A and that A/I is a topological ∗-algebra with respect to the quotient topology τq . Let φ denote the quotient ∗-homomorphism of A onto A/I . Then, φ A[B0 ] is a ∗homomorphism of A[B0 ] into A/I , with the kernel being the norm closed two-sided ideal I0 of A[B0 ]. Consequently, φ(A[B0 ]), equipped with the quotient norm φ(x)q = inf x + y, x ∈ A[B0 ], y∈I0

is ∗-isomorphic to A[B0 ]/I0 , therefore a C ∗ -algebra. Since x(e + x ∗ x)−1 ∈ A[B0 ], for all x ∈ A, as we noticed above, it follows that

−1 (x + I ) (e + I ) + (x + I )∗ (x + I ) = x(e + x ∗ x)−1 + I ∈ φ(A[B0 ]) for all x ∈ A. Since the relative topology on φ(A[B0 ]) of the topology τq is weaker than the ·-topology on φ(A[B0 ]), and since the norm closed unit ball of φ(A[B0 ]) is bounded with respect to the norm topology on φ(A[B0 ]), it is immediate that the norm closed unit ball of φ(A[B0 ]) is τq -bounded. Hence, by Theorem 7.4.5(ii), (A/I )[τq ] is a GB∗ -algebra over the C ∗ -algebra A[B0 ]/(I ∩ A[B0 ]).   An interesting application of Theorem 7.4.8 is the following, which clearly extends Corollary 3.3.11(i). Theorem 7.4.9 (Kunze) If A[τ ] is a complete m∗ -convex algebra with an identity element e, which is also a GB∗ -algebra, then A[τ ] is a pro-C ∗ -algebra. Proof For each p ∈ , the set Np = {x ∈ A : p(x) = 0} is a closed *-ideal of A, therefore by Theorem 7.4.8, we obtain that (A/Np )[τq ] is a GB∗ -algebra (with respect to the quotient topology τq ). On the other hand, A/Np is a normed ∗algebra with respect to the topology defined by the norm x + Np p := p(x), x ∈ A. Since (A/Np )[τq ] is a GB∗ -algebra, it follows that (A/Np )[B0 ] is a C ∗ -algebra with respect to  · B0 , where B0 is now the greatest member in B∗(A/Np )[τq ] . But, (A/Np )[ · p ] is also a ∗-normed algebra and moreover “A/Np is Allan symmetric”, being a GB∗ -algebra. So, it follows from Proposition 7.4.1 that A/Np = (A/Np )[B0 ], as algebras. Let B = (A/Np )[B0 ]. Then U (B) = B0 is  · p -bounded. This is due to the fact that B0 is τq -bounded, hence  · p -bounded, because  · p ≺ τq . Therefore, by Theorem 7.4.5, (A/Np )[ · p ] is a GB∗ -algebra. Hence, by Corollary 7.4.2, A/Np is a C ∗ -algebra with respect to some norm  ·  p equivalent to  · p . It follows that the topology τ can be defined by a family

= {p } of C ∗ -seminorms with p (x) := x + Np  p , x ∈ A, so that A becomes a pro-C ∗-algebra, under the equivalent to τ topology τ determined by the family

of C ∗ -seminorms as before.   Notes All results in this section are due to W. Kunze and can be found in [98].

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7.5 A Vidav-Palmer Theorem for GB*-Algebras The well known Vidav–Palmer theorem provides us with a concrete characterization of C ∗ -algebras as those complex Banach algebras with identity, which are spanned, as linear spaces, by elements with real numerical range (for the definition of the numerical range of an element in a normed algebra see Remark 7.5.5). In this section we present various types of Vidav–Palmer Theorem in the context of locally convex ∗-algebras (among them being GB∗ -algebras), all of them credited to A.W. Wood [156]. Theorem 7.5.1 (Vidav–Palmer) Let A[ · ] be a Banach algebra with identity, such that A = H + iH, where H denotes the set of elements with real numerical range. Then A[ · ] is isometrically isomorphic with a C ∗ -algebra. In [156], A.W. Wood explores a possible generalization of the aforementioned result in the realm of GB∗ -algebras. In order to do so, he defines a natural generalization of the notion of the numerical range, in the setting of a locally convex algebra (see Definition 7.5.2). Recall that given a locally convex space E[τ ], the symbol E denotes its topological dual and = {p} a family of seminorms defining the topology τ . In the sequel, stands for the family of seminorms   qB (f ) = sup |f (x)| : x ∈ B , B ∈ B, f ∈ E , where B denotes the collection of τ -bounded subsets of E, of the form   x ∈ E : p(x) ≤ Mp , p ∈ , Mp > 0 . The family {qB }B∈B defines the topology β(E , E) on E , i.e., the topology of uniform convergence on weakly bounded subsets of E. The latter implication follows from the fact that the τ -bounded and the weakly bounded subsets of E coincide (see [128, p. 67, Theorem 1]).

 In the rest of this section, we shall use the notation (E, ), (A, ), for a given locally convex space E[τ ], respectively a given locally convex algebra A[τ ], whose topology τ is defined by a family of seminorms = {p}. Definition 7.5.2 Let (E, ) be a locally convex space. Then, for every p ∈ , let   p := (x, f ) ∈ E × E : p(x) = f (x) = 1, |f (y)| ≤ p(y), ∀ y ∈ E .

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For T ∈ L(E), where L(E) is the set of all linear continuous operators on the space E, let    Wp1 (T ) := f (T x) : (x, f ) ∈ p and W 1 (T ) := Wp1 (T ). p∈

Likewise, if E

is the double topological dual of (E, ), let q ≡ qB ∈ and   q := (f, x

) ∈ E × E

: x

(f ) = 1 = q(f ), |x

(g)| ≤ q(g), ∀ g ∈ E

and   Wq1 (T ), Wq1 (T ) := x

(T f ) : (f, x

) ∈ q }, W 2 (T ) := q∈

where T denotes the transpose of T . Then, W (T ) := W 1 (T )



W 2 (T ) is called the numerical range of T .

For simplicity we shall use the notation W (T ). Remark 7.5.3 (1) The previous definition is a generalization of the respective definition of the spatial numerical range of an operator T ∈ L(E) for a normed space E. We recall that in the latter case, the spatial numerical range of T ∈ L(E) is denoted by V (T ) and is the set   V (T ) := f (T x) : (x, f ) ∈  , where

  :={(x, f ) ∈ E × E : x = |f (x)| = f  = 1 . Clearly, V (T ) coincides with W 1 (T ) of Definition 7.5.2. In addition, when E is a Banach space, by [31, Corollary 17.3] we have that V (T ) ⊂ V (T ) ⊂ V (T ). Therefore, since V (T ) coincides with W 2 (T ) of Definition 7.5.2, we have that for a Banach space E and T ∈ L(E), the sets W 1 (T ) and W 2 (T ) have the same closure. (2) It is apparent from Definition 7.5.2 that the numerical range of T ∈ L(E) depends on the family of seminorms defining the topology of the locally convex space E. Based on Definition 7.5.2, the following definition describes the numerical range of an element in an abstract locally convex algebra. Definition 7.5.4 Let (A, ) be a locally convex algebra and let a ∈ A. The numerical range of a, denoted by W (a) (or simply by W (a)), is the set W (Ta )

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7 Applications I: Miscellanea

(or simply the set W (Ta ) as noticed above), where Ta ∈ L(A) is the linear operator Ta x := ax, x ∈ A. An element h ∈ A, such that W (h) ⊂ R, is said to be hermitian. The set of all hermitian elements of A is denoted by H (or by H if no confusion arises). Remark 7.5.5 In case A is a Banach algebra with identity, the classical definition of the numerical range of an element a ∈ A is that given by the set   V (A, a) := f (ax) : x ∈ S(A), f ∈ D(A, x) ,   where S(A) := x ∈ A : x = 1 and   D(A, x) = f ∈ A : f (x) = 1 = f  (more generally, this is the definition given for the numerical range of an element of a normed algebra with identity; see [30, Definition 2.1]). We note that V (A, a) coincides with W 1 (Ta ) of Definition 7.5.4. So, V (A, a) = W 1 (Ta ) ⊂ W (a). On the other hand, for λ ∈ W 1 (Ta ), we have that λ ∈ W 1 (Ta ) = W 1 (Ta ), where the equality is based on [31, Corollary 17.3]. Thus, since W 1 (Ta ) is closed (see [30, §2, Theorem 3]), we have that λ ∈ W 1 (Ta ), i.e., W 1 (Ta ) ⊂ V (A, a). Therefore, we conclude that in the case of a Banach algebra, the definition of the numerical range of an element of the algebra, as it is given in Definition 7.5.4, coincides with the classical definition. For a hypocontinuous locally convex algebra, the set H of hermitian elements is closed, as it is implied by Proposition 7.5.6, below. We recall that a locally convex algebra A[τ ] is hypocontinuous, if for every bounded subset B of A[τ ] and each 0-neighbourhood U there is a 0-neighbourhood V , such that BV ⊂ U and V B ⊂ U . Note that in a hypocontinuous topological algebra the product of two bounded subsets is bounded. Indeed, let A[τ ] be a hypocontinuous locally convex algebra and let B1 , B2 be two bounded subsets of A[τ ]. Let U be a 0-neighbourhood in A[τ ]. Then, there is a 0-neighbourhood V and a positive number λ, such that B1 ⊂ λV and V B2 ⊂ U, B2 V ⊂ U. Hence, B1 B2 ⊂ λV B2 ⊂ λU . Therefore, B1 B2 is bounded. Proposition 7.5.6 Let (A, ) be a hypocontinuous locally convex algebra and let (aλ )λ∈ be a net in A, such that aλ → a ∈ A. If K is a closed subset of C, such that W (aλ ) ⊂ K, for all λ ∈ , then W (a) ⊂ K. Proof Let p ∈ and (x, f ) ∈ p . Then f (aλ x) ∈ W (aλ ) and so f (aλ x) ∈ K, for all λ ∈ . Since the multiplication in A is separately continuous and K is closed, we have that f (ax) ∈ K. Thus, W 1 (Ta ) ⊂ K. Next, we are going to show that W 2 (Ta ) ⊂ K. Let (f, x

) ∈ q , for some q ≡ qB ∈ . We show that Ta λ f → Tα f with respect to β(A , A) : let ε > 0

7.5 A Vidav-Palmer Theorem for GB*-Algebras

181

and B a σ (A, A )-bounded subset of A. By the   comments before Definition 7.5.2, it suffices to take B ≡ x ∈ A : p(x) ≤ Mp , p ∈ , where {Mp : p ∈ } is any family of positive numbers. Clearly the set B is then a bounded subset of the locally convex algebra (A, ). By continuity of f ∈ A , there is a seminorm p ∈ , such that |f (x)| ≤ p(x), for all x ∈ A. If we consider the 0-neighbourhood U = y ∈  A : p(y) < 2ε of A, due to hypocontinuity of A there is a 0-neighbourhood V , such that BV ⊂ U and V B ⊂ U . Since aλ → a, there is λ0 ∈ , with aλ − a in V , for all λ ≥ λ0 . Therefore, 

 ε

|f (aλ − a)x | ≤ p (aλ − a)x < , ∀ λ ≥ λ0 and x ∈ B. 2 Observe that, there is some x0 ∈ B, such that 

  

ε sup |f (aλ − a)x | : x ∈ B < |f (aλ − a)x0 | + . 2   Hence, sup |f ((aλ − a)x)| : x ∈ B < ε, for all λ ≥ λ0 . So,   sup |(Ta λ − Ta )f (x)| : x ∈ B → 0, i.e., Ta λ f → Ta f, with respect to β(A , A).

Consequently, x

(Ta λ f ) → x

(Ta f ), from which it follows that x

(Tα f ) ∈ K. Hence, we derive the desired inclusion W 2 (Ta ) ⊂ K and that W (a) = W 1 (Ta ) ∪ W 2 (Ta ) ⊂ K.

 

Definitions 7.5.2 and 7.5.4 suggest that an element of a locally convex algebra A[τ ] can interchangeably be viewed, in terms of its numerical range, as a continuous operator in the algebra L(A). The next definition distinguishes a particular bounded subset of L(E), where E is a general locally convex space, which holds an important role in the course of obtaining a generalized Vidav–Palmer theorem for locally convex algebras (in this respect, see, for instance, Theorem 7.5.42). Definition 7.5.7 Let (E, ) be a locally convex space. The set of -contractions, denoted by C , is the set   C = T ∈ L(E) : p(T x) ≤ p(x), x ∈ E, p ∈ . It is clear that C is an absolutely convex, bounded, closed subset of L(E) with respect to the topology of uniform convergence on bounded subsets of E. Moreover it is straightforward that C 2 ⊂ C and I dE ∈ C , where I dE denotes the identity map on E.  Then, A[C  ], the subalgebra of L(E) generated by C , equals to λT : λ ∈ C, T ∈ C and it is a normed algebra, with respect to the norm   T  = inf μ > 0 : T ∈ μC , T ∈ A[C ].

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7 Applications I: Miscellanea

We note that   T  = sup p(T x) : p ∈ , x ∈ E, p(x) ≤ 1 . Indeed, if T ∈ λC , for some λ > 0, then T = λQ, Q ∈ C . Hence,   sup p(T x) : p ∈ , x ∈ E, p(x) ≤ 1 ≤   λ sup p(Qx) : p ∈ , x ∈ E, p(x) ≤ 1 ≤ λ, thus     sup p(T x) : p ∈ , x ∈ E, p(x) ≤ 1 ≤ inf λ > 0 : T ∈ λC .   On the other hand, if sup p(T x) : p ∈ , x ∈ E, p(x) ≤ 1 ≡ s, then for every ε > 0 we have that p

 1   1 T x = p(T x) ≤ p(x). s+ε s+ε

The previous inequality is certainly true for x ∈ E with p(x) = 0. For x ∈ E with p(x) = 0, if T = μQ, where μ ∈ C, Q ∈ C , we have that p(T x) = |μ|p(Qx) ≤ |μ|p(x) = 0, 

  1 thus the aforementioned inequality is still valid. So, by p s+ε T x ≤ p(x), we   1 have that s+ε T ∈ C , thus inf λ > 0 : T ∈ λC ≤ s + ε. Since ε > 0 is arbitrary   we obtain inf λ > 0 : T ∈ λC ≤ s. Furthermore, by considerations similar to those of the paragraph preceding Definition 2.2.3, the topology of uniform convergence on bounded subsets of E, which A[C ] carries as a subalgebra of L(E), is weaker than the norm-topology  ·  . If (E, ) is a locally convex space and T ∈ A[C ], the numerical range of T regarded as an element of the normed algebra A[C ] with identity, is denoted by V (A[C ], T ) (or V (A[C ], T )). By [30, Lemma 2.2], the numerical range V (A[C ], T ) of an element T ∈ A[C ], coincides with the set {f (T ) : f ∈ D(A[C ], I d)}, where D(A[C ], I d) stands for the set of all continuous linear functionals f : A[C ] → C, such that f  = 1 = f (I d) and I d denotes the identity map on A[C ]. For an element T ∈ A[C ], the relation between the sets V (A[C ], T ) and W (T ) is given by the following result. Proposition 7.5.8 Let (E, ) be a locally convex space and let T ∈ A[C ]. Then, W (T ) ⊂ V (A[C ], T ).

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183

Proof Let μ ∈ W 1 (T ). Then, μ ∈ Wp1 (T ), for some p ∈ , i.e., μ = f (T x), for some (x, f ) ∈ p . Consider the map g : A[C ] → C : S → g(S) := f (Sx), (x, f ) ∈ p . For every S ∈ A[C ], we have that |g(S)| = |f (Sx)| ≤ p(Sx) ≤ S p(x) = S . So, g ≤ 1 and since g(I d) = 1 = f (x), we have that g ∈ D(A[C ], I d). Therefore, f (T x) = g(T ) ∈ V (A[C ], T ), which proves the inclusion of W 1 (T ) in V (A[C ], T ). In order to obtain the inclusion W 2 (T ) ⊂ V (A[C ], T ), we first note that T  = T  , for every T ∈ A[C  ]. The latter relation holds true, since for F ∈ C , for a bounded set B = x ∈ E : p(x) ≤ Mp , p ∈ , where Mp > 0, p ∈ , and for every f ∈ E , we have that     qB (F f ) = sup |(F f )(x)| : x ∈ B = sup |f (F x)| : x ∈ B   ≤ sup |f (x)| : x ∈ B = qB (f ), where the inequality is based on the inclusion F (B) ⊂ B, as can be easily seen by the fact that F ∈ C . Hence, F ∈ C . Therefore, for T ∈ A[C ], TT ∈ C and

thus TT ∈ C , so T  ≤ T  . By symmetrical arguments, T  ≤ T  . Let μ ∈ W 2 (T ). Then, μ ∈ Wq1 (T ) for some q ∈ . Thus, μ = x

(T f ), for some (f, x

) ∈ q . Consider the map h : A[C ] → C : S → h(S) := x

(S f ). Then, |h(S)| = |x

(S f )| ≤ q(S f ) ≤ S  q(f ) = S q(f ). So, h = 1 = h(I d), hence, μ = x

(T f ) = h(T ) ∈ V (A[C ], T ). Therefore, we conclude that W (T ) = W 1 (T ) ∪ W 2 (T ) ⊂ V (A[C ], T ).

 

Proposition 7.5.9 Let E[τ ] be a locally convex space. If B is a subset of L(E), such that B 2 ⊂ B, then B is equicontinuous, if and only if, there is a defining family of seminorms 

for τ , equivalent to the given one , such that B ⊂ C

. Proof ⇐ Let 

be a defining family of seminorms for τ , equivalent to the respective given family (for τ ), such that B ⊂ C

. Then, it is straightforward that T (V ) ⊂ V , for every 0-neighbourhood V , with respect to 

and every T ∈ B. Hence, B is equicontinuous, with respect to 

, hence with respect to , too. ⇒ Let B be equicontinuous, with respect to = {p}. Consider,   p (x) := sup p(T x) : T ∈ B , x ∈ E.

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7 Applications I: Miscellanea

Then, the seminorm p  is well-defined and this is a straightforward consequence of   the equicontinuity of B, with respect to

. The family of seminorms

= p  : p ∈ 

defines the same topology on E as the topology defined on E by the family . For this, note that on the one hand, we can assume without loss of generality, that I dE ∈ B. Then, if (xλ )λ∈ is a net in E, such that xλ → 0, it follows immediately { p}

that xλ → 0. On the other hand, if xλ → 0, then for every T ∈ B, T xλ → 0. So, {p}

{p}

for ε > 0 there exist λ0 ∈  and T0 ∈ B, such that

{p}

ε 

  sup p T xλ : T ∈ B < + p T0 xλ < ε, ∀ λ ≥ λ0 ; hence, xλ → 0. { p} 2 Moreover, for T ∈ B we have that 

    p (T x) = sup p S(T x) : S ∈ B ≤ sup p(Rx) : R ∈ B = p (x), for every x ∈ E, p ∈ , where the inequality is due to the property B 2 ⊂ B. Therefore, T ∈ C  

, thus B ⊂ C

. Lemma 7.5.10 Let A[τ ] be a hypocontinuous locally convex algebra and B a bounded subset of A, such that B 2 ⊂ B. Then, there is a defining family of seminorms 

for τ , equivalent to the given one , such that {Ta : a ∈ B} ⊂ C

(for Ta , a ∈ A, see Definition 7.5.4). Proof Let B = {Ta : a ∈ B}. For every b, c ∈ B, Tb Tc = Tbc , where bc ∈ B 2 ⊂ B. So, (B )2 ⊂ B . Moreover, by hypocontinuity of A[τ ] and the assumption for B being bounded, we have that for any given 0-neighbourhood V in A[τ ], there is a 0-neighbourhood U in A[τ ], such that BU ⊂ V . The previous inclusion implies that B is equicontinuous. Then, by Proposition 7.5.9, there is a defining family of seminorms 

for τ , equivalent to the given one , such that B ⊂ C  

. Remark 7.5.11 Recall that the map A → L(A) : a → Ta , with Ta (x) := ax, for all x in A, is injective. So, in what follows, for simplicity of notation, we find it convenient to identify an element of a locally convex algebra A[τ ] with its left regular representation a ∈ A → Ta ∈ L(A) and write a instead of Ta . Moreover, we choose to write a in place of Ta  , for a ∈ A with Ta ∈ A[C ]. Observe that by this natural identification of a with Ta there are two relevant spectra of a ∈ A. Namely,     / A0 σA (a) = λ ∈ C : λe − a has no inverse in A0 ∪ ∞, if and only if, a ∈   and σC (a) = λ ∈ C : λI dA − Ta has no inverse in A[C ] ∪   ∞, if and only if, Ta ∈ / A[C ] . The following lemma shows that these two sets coincide for a hypocontinuous locally convex algebra. Hence, for a hypocontinuous locally convex algebra, the

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185

aforementioned identifications of a with Ta would not lead to any ambiguity with respect to the spectrum of a. Lemma 7.5.12 Let (A, ) be a hypocontinuous locally convex algebra with identity e and a ∈ A. Then, σA (a) = σC (a). Proof We begin by establishing that ∞ ∈ σC (a) if and only if ∞ ∈ σA (a). Let ∞ ∈ σA (a). Then, a ∈ / A0 . With the aim to reach a contradiction suppose /

 that ∞ ∈ σC (a). Hence, Ta ∈ A[C ]. So, there exists k > 0, such that p Ta x ≤ kp(x), 

for every x ∈ A, p ∈ . Therefore, by induction we have that p ( 1k a)n ≤ 1, i.e., a ∈ A0 , a contradiction. On the other hand let ∞ ∈ σC (a) and suppose that ∞ ∈ / σA (a). Then, there is λ > 0 such that the set B = {(λa)n : n ∈ N} is bounded. Consider F = B ∪ {e}. Clearly, F is bounded and F 2 ⊂ F . By Lemma 7.5.10 there is a defining family of seminorms for the topology of A, say 

, such that {Tb : b ∈ F } ⊂ C

. We recall that by the proof of Proposition 7.5.9 the family 

is given by the seminorms 

  p (x) = sup p Tb x : b ∈ F , p ∈ . Hence, by the very definitions, the form of p  and the fact that Tλa ∈ C

, we deduce Tλa  ≤ 1, so Ta ∈ A[C ]. This last fact contradicts our assumption that ∞ ∈ σC (a) in the beginning of this paragraph. Let now λ ∈ σA (a) ∩ C. We want to show that λ ∈ σC (a). Supposing that λ∈ / σC (a) we have −1 −1 −1 (λI d − Ta )−1 = Tλe−a ∈ A[C ], so that Tλe−a Tλe−a = Tλe−a Tλe−a = I d. −1 By this last equality we conclude that (λe−a)−1 exists and that T(λe−a)−1 = Tλe−a ∈ −1 −1 / σC ((λe − a) ). Therefore, ∞ ∈ / σA ((λe − a) ), i.e., (λe − A[C ]. So, ∞ ∈ a)−1 ∈ A0 , a contradiction. Finally, by similar to the considerations of the previous paragraph one can show that

σC (a) ∩ C ⊂ σA (a), a ∈ A.

 

It should be noted that the spectrum σC (a), a in A, as this is defined in Remark 7.5.11, is a slight generalization of the spectrum denoted by σA[C ] (a) in Proposition 2.3.12.

 For simplicity of notation, in what follows throughout the section, we write σ (a) for σC (a). Theorem 7.5.14 reveals a relation between the spectrum σ (a) and the numerical range W (a) of an element a of a complete (continued)

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7 Applications I: Miscellanea

locally convex algebra. The following result can be seen as a step towards Theorem 7.5.14. Theorem 7.5.13 Let (E, ) be a complete locally convex space and T ∈ L(E). If λ ∈ C \ W (T ), then (λI dE − T )−1 exists and     (λI dE − T )−1  ≤ 1 , with k := inf |λ − μ| : μ ∈ W (T ) .

k Proof Let (x, f ) ∈ p , p ∈ . Then (see Definition 7.5.2), ! ! ! ! !  !

p (λI dE − T )x ≥ !f (λI dE − T )x ! = !λf (x) − f (T x)! = !λ − f (T x)! ≥ k. Therefore, we have that

 p (λI dE − T )x ≥ kp(x), for all p ∈ , x ∈ E.

(7.5.4)

Moreover, for (h, x

) ∈ q , q ∈ (ibid.), we have that !

 !

! ! q (λI dE − T ) h ≥ !x

(λI dE − T ) h ! = !x

(λI dE − T )h! ! ! ! ! = !λx

(h) − x

(T h)! = !λ − x

(T h)! ≥ k. Therefore,

 q (λI dE − T ) h ≥ kq(h), for all q ∈ , h ∈ E .

(7.5.5)

By (7.5.4) we have that λI dE − T is injective and that (λI dE − T )E is closed in E. Furthermore, by (7.5.5), (λI dE − T )E is dense in E: indeed if (λI dE − T )E = E, then by the Hahn–Banach theorem, there is an f ∈ E , such that f = 0 and f (λI dE −T )E = 0. Therefore, (λI dE − T ) f (E) = 0 with f = 0, which contradicts (7.5.5). Hence, λI dE − T is onto E. Hence, (λI dE − T )−1 exists and applying (7.5.4) for the element (λI dE − T )−1 x, in the place of x we conclude that

 1 p (λI dE − T )−1 x ≤ p(x), ∀ p ∈ , x ∈ E. k 

−1   ≤ 1. Consequently, (λI dE − T )−1 ∈ A[C ] and  λI dE − T k

  ∗

Theorem 7.5.14 Let (A, ) be a complete locally convex algebra and let W (a) ∗ ∗ denote the if λ ∈ C and  closure of W (a) inC . Then, σ (a) ⊂ W−1(a) . Moreover, k := inf |λ − μ| : μ ∈ W (a) > 0, then (λe − a)  ≤ 1k .

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Proof For λ ∈ C, the result is a direct consequence of Theorem 7.5.13 and the comments just before it. Hence, what is left to be shown is that for ∞ ∈ σ (a); ∗ we have ∞ ∈ W (a) . Equivalently, it suffices to show that if W (a) is bounded, then a ∈ A[C ]. Towards this direction, let M > 0, such that |μ| ≤ M, for all μ ∈ W (a). Then, for every |λ| > M, one has λ ∈ / W (a). So, by Theorem 7.5.13, (λe − a)−1 exists and     −1 (λe − a)−1  ≤ inf |λ − z| : z ∈ W (a) ≤

1 . |λ| − M

Then, following the proof of Theorem 2.3.7, we get that the map λ → (λe − a)−1 is analytic at ∞. Therefore, by Theorem 2.3.7(1), ∞ ∈ / σ (a), hence a ∈ A[C ].   For a pseudo-complete locally convex algebra A, the algebra A[C ] is, by the definition of pseudo-completeness, a Banach algebra. Therefore, for a ∈ A[C ] we can define exp(a) ∈ A[C ] by means of the usual power series, i.e., exp(a) =  ∞ an n=0 n! , where the series converges with respect to  ·  . The use of the exponential function will give us, at a first stage, a closer connection between W (a) and V (A[C ], a) (see Proposition 7.5.19) than that we have already obtained at Proposition 7.5.8. First we need the following definition and a preparatory result. Definition 7.5.15 Let (A, ) be a locally convex algebra. (i) An element a ∈ A is said to be dissipative if Reλ ≤ 0, for every λ ∈ W (a). (ii) We say that the resolvent of a ∈ A has first order decay in the right halfplane, if for λ ∈ C and Reλ > 0, then the element (λe − a)−1 exists and  (λe − a)−1  ≤ 1 . Reλ

Remark 7.5.16 We observe that if a is a dissipative element of a complete locally convex algebra (A, ), then the resolvent of a has first order decay in the right half∗ plane. Indeed, if λ ∈ C and Reλ > 0, then λ ∈ / W (a) . Hence, by Theorem 7.5.14, we have that   (λe − a)−1  ≤

1 1 ≤ , inf{|λ − μ| : μ ∈ W (a)} Reλ

where the second inequality follows from the fact that Reλ ≤ Reλ − Reμ, for every μ ∈ W (a). As we shall see in Proposition 7.5.21, the converse of the implication of the previous paragraph holds for a pseudo-complete and hypocontinuous locally convex algebra. Proposition 7.5.17 Let (A, ) be a pseudo-complete locally convex algebra and a ∈ A[C ], such that its resolvent has first order decay in the right half-plane. Then,  exp(a) ≤ 1. Proof Consider an = n2 (ne − a)−1 − ne, for n ∈ N. Note that (ne − a)−1 exists due to the assumption of the resolvent of a having first order decay in the right

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7 Applications I: Miscellanea

half-plane and moreover ne − a ≤ n1 . Then, we have that     an − a  = n2 (ne − a)−1 − ne − a 



  −1 2  = (ne − a) (n e − (ne − a)(ne + a))

  = a 2 (ne − a)−1 

≤

 1 a 2  → 0.

n

Hence, an → a with respect to  ·  . Then, n   

 (an − a)k    exp an − a − e = lim   

n→∞

k! k=1

≤ lim

n→∞

n 

  an − a k

k=1

k!

  = exp(an − a  ) − 1 → 0. Therefore,      exp(an ) − exp(a) =  exp(an − a + a) − exp(a)



    = (exp(an − a) − e) exp(a)

    ≤  exp(an − a) − e  exp(a) → 0, where the second equality is due to the fact that an and a commute. Moreover, we have that      exp(an ) =  exp(n2 (ne − a)−1 − ne)



  2 −1 =  exp(n (ne − a) ) exp(−ne)

  ≤ e−n exp n2 (ne − a)−1 

≤e   Therefore,  exp(a) ≤ 1.

n(n(ne−a)−1  −1)

≤ 1.  

The following Proposition 7.5.19 provides us with a clear view of the way the sets W (a), V (A[C ], a) fit together, where (A, ) is a compete locally convex algebra and a is an element of A[C ]. Essential to the proof of Proposition 7.5.19 is the Lumer–Phillips theorem for Banach algebras with identity, which is subsequently stated. For the proof of the latter result the reader is referred to [30, Theorem 3.6].

7.5 A Vidav-Palmer Theorem for GB*-Algebras

189

Theorem 7.5.18 (Lumer–Phillips)  Let A[ · ]  be a Banach algebra with identity and let a ∈ A. Then, V (A, a) ⊂ λ : Reλ ≤ 0 , if and only if,  exp(ta) ≤ 1, for all t ≥ 0. Proposition 7.5.19 Let (A, ) be a complete locally convex algebra and let a ∈ A[C ]. Then, V (A[C ], a) ⊂ co W (a) (the latter indicates the closed convex hull of W (a)). Proof If we suppose that W (a) ⊂ {λ : Reλ ≤ 0}, then for every t > 0 the element ta is dissipative. Then, by Remark 7.5.16, we deduce that ta has resolvent with first order decay in the right half-plane. Therefore, by Proposition 7.5.17,  exp(ta) ≤ 1. Then, it follows by Theorem 7.5.18 that V (A[C ], a) ⊂ {λ : Reλ ≤ 0}. Aiming to obtain a contradiction, suppose that there exists λ ∈ V (A[C ], a), such that λ ∈ / co W (a). Then, {λ} and co W (a) are disjoint closed convex subsets of C. Thus, by the Hahn–Banach separation theorem (see, for instance, [36, Theorem 1.7, p. 7]), there is a linear functional x ∗ : C → R and α ∈ R, such that       λ ⊂ μ ∈ C : x ∗ (μ) > α and co W (a) ⊂ μ ∈ C : x ∗ (μ) < α . (7.5.6) For x ∗ , there exist α1 , α2 ∈ R, such that x ∗ (μ) = α1 Re(μ) + α2 Im(μ), for all μ in C. Therefore, we have that     W (a) ⊂ co W (a) ⊂ μ ∈ C : x ∗ (μ) < α ⊂ μ ∈ C : α1 Re(μ)+α2 Im(μ) < α . Let α1 = 0. Then, after some calculations we have that   W (a) ⊂ μ ∈ C : α1 Re(μ) − α + α2 Im(μ) < 0    α α + α2 Im μ − 0 (see quotation at the beginning of the last paragraph). Consider the set C = {dt : t ∈ R} · B, i.e., the product of the two sets.   Since A is hypocontinuous, C is bounded. Hence, if V = g ∈ A : qC (g) < ε , then for every t ∈ R and g ∈ V , we have that   qB (dt g) = sup |dt g(x)| : x ∈ B   = sup |g(dt x)| : x ∈ B   ≤ sup |g(y)| : y ∈ C < ε. So, dt V ⊂ U for every t ∈ R, i.e., {dt : t ∈ R} is equicontinuous. Therefore, from (7.5.12) we have that 

  1

2

q (e + t (hk − kh) )f − 1 ≤ t qB dt f , B t2 so that lim sup t →0+

  

1

2

+ t (hk − kh) )f − 1 ≤ lim sup t qB dt f = 0, q (e B 2 t t →0+

where the last equality follows since {dt : t ∈ R} is equicontinuous.

7.5 A Vidav-Palmer Theorem for GB*-Algebras

195

Then, for every (f, x

) ∈ q , we have 

   1 

Re x

(hk − kh) f = Re x

(e + t (hk − kh) )f − 1 t  ! 1 !!



x (e + t (hk − kh) )f ! − 1 ≤ t   1

≤ qB (e + t (hk − kh) )f − 1 . t So, we deduce W 2 (hk − kh) ⊂ iR.

(7.5.13)

Thus, relations (7.5.11) and (7.5.13), show that i(hk − kh) ∈ H.

 

Theorem 7.5.26 provides a generalization of a result, which is known in the theory of numerical ranges of elements in normed algebras as Vidav’s lemma (see [30, Theorem 5.10]). The proof of Theorem 7.5.26 is based on the following complex analytical result, whose the statement we include for convenience of the reader and for its proof we refer to [156, Proposition 6.2]. Proposition 7.5.25 Let g be a complex-valued analytic function on {z ∈ C : Rez > 0}, continuous on the closure in C of the latter set. Suppose that for z ∈ / R, |g(z)| ≤ 1 1 . Then, |g(z)| ≤ , for Rez > 0. |Imz| |z| Theorem 7.5.26 Let (A, ) be a hypocontinuous complete locally convex algebra

and let h ∈ A be hermitian. Then, sup σ (h) ∩ R = sup W (h). Proof We first show that if σ (h) ∩ R ⊂ (−∞, 0), then W (h) ⊂ (−∞, 0]. For this suppose that σ (h) ∩ R ⊂ (−∞, 0) and let f ∈ A , such that |f (a)| ≤ 1, for a ∈ A[C ]. For z ∈ C, such that Rez ≥ 0, consider the function g(z) = f (ze − h)−1 . It follows from Theorem 2.3.7 that the function g is analytic on {z ∈ C : Rez ≥ 0}. Moreover, since h is hermitian, we have that W (ih) ⊂ {z ∈ C : Rez = 0} and thus ih is dissipative. Hence, by Remark 7.5.16, ih has resolvent of first order decay in the right half-plane. Let z ∈ / R, such that Imz < 0. Then, ! ! ! !   !g(z)! = !f (ze − h)−1 ! ≤ (ze − h)−1  ≤

1 1 = . Re(iz) −Imz

1 Similarly, we deduce that |g(z)| ≤ Imz , in case Imz > 0. Therefore, the function g satisfies the conditions of Proposition 7.5.25, hence 1 |g(z)| ≤ |z| , for z ∈ C, such that Rez > 0. By the Hahn–Banach theorem, it follows

1 , for z ∈ C, such that Rez > 0. That is, the resolvent that (ze − h)−1  ≤ |z| of h has first order decay in the right half-plane. Then, by Proposition 7.5.21, h is dissipative and so W (h) ⊂ (−∞, 0].

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7 Applications I: Miscellanea

Let λ = sup(σ (h) ∩ R) and μ = sup W (h). By Theorem 7.5.14, we have λ ≤ μ. Suppose that μ > λ. Since λ = sup(σ (h) ∩ R) = sup(co(σ (h) ∩ R)), by the Hahn–Banach separation theorem we have that there are α, α1 ∈ R, in such a way that

       co σ (h) ∩ R ⊂ x ∈ R : α1 x < α and μ ⊂ x ∈ R : α1 x > α . Then,

  

 σ α1 h − αe ∩ R = α1 λ0 − α : λ0 ∈ σ (h) ∩ R ⊂ − ∞, 0 . Therefore, W (α1 h − αe) ⊂ (−∞, 0]. Equivalently, α1 W (h) ⊂ (−∞, α], a contradiction.   In order to build up to a Vidav–Palmer type theorem for a GB∗ -algebra A[τ ], A. Wood proposed in [156, p. 260] a slight variant of the family of subsets of the given locally convex ∗-algebra than that of Allan’s collection B∗A (see Definition 3.3.1). In fact, for a locally convex algebra A[τ ] with identity e and with an involution ∗,  ∗ of subsets of A, such that A. Wood introduces the family B A  ∗ is bounded and absolutely convex; (1) each B ∈ B A (2) e ∈ B, B 2 ⊂ B and B = B ∗ .  ∗ and the family B∗ of Notice that the only difference between the family B A A ∗  are not assumed to be closed. Definition 3.3.1 is that the sets of the family B A The reason behind this choice, in this collection of subsets in A[τ ], can be spotted in the fact that, in Wood’s course of arguments in [156], the continuity of the involution is not taken for granted (in this respect, see, for instance, the proof of Proposition 7.5.33, in which the assumption of closedness would impede us from having the same greatest member for the two topologies in question). Subsequently, the presence or absence of the assumption of continuity of the involution in a locally convex algebra provides us with certain variations in the respective definitions of a GB∗ -algebra. In concrete terms, we have the following Definition 7.5.27 A semi-HB∗ algebra is a locally convex algebra A[τ ] with identity e and with an involution (not necessarily continuous), such that  ∗ has a greatest member B0 , which is τ -closed and A[B0 ] is a Banach ∗(1) B A algebra with respect to  · B0 . (2) (e + h2 )−1 ∈ A0 , for every h ∈ H (A), i.e., for every h ∈ A, such that h∗ = h. In case the involution is symmetric (see Definition 3.1.6), A[τ ] is said to be a semi-GB∗ -algebra. A semi-GB∗ -algebra with a continuous involution is called a GB∗ -algebra, while an HB∗ -algebra is a semi-HB∗ -algebra with a continuous involution. The previous definition of a GB∗ -algebra is equivalent to the definition of a GB∗ algebra of Dixon, as the latter is given in Definition 3.3.5. Indeed, on the one hand

7.5 A Vidav-Palmer Theorem for GB*-Algebras

197

let A[τ ] be a GB∗ -algebra according to Definition 3.3.5 and let B0 be the greatest  ∗ . Then, the τ -closure B of B belongs to B∗ , member of B∗A . Consider B ∈ B A A ∗ ∗  , we have that B0 is also the greatest member of hence B ⊂ B0 . Since BA ⊂ B A  ∗ . Therefore, it follows that A[τ ] is a GB∗ -algebra according to Definition 7.5.27. B A On the other hand, let A[τ ] be a GB∗ -algebra according to Definition 7.5.27 and  ∗ . Consider B ∈ B∗ . Then, B ∈ B ∗ , let B0 be the greatest member of the family B A A A hence B ⊂ B0 . Since by Definition 7.5.27 B0 is supposed to be τ -closed, we have that B0 ∈ B∗A . So, B0 is the greatest member of B∗A , from which it follows that A[τ ] is also a GB∗ -algebra according to Definition 3.3.5. Remark 7.5.28 We recall that, by Definition 3.1.3, a locally convex algebra A with involution is hermitian if σA (h) ⊂ R, for every h ∈ H (A). If A is a pseudocomplete locally convex algebra with identity e, then from Proposition 3.1.5 we have that for h ∈ H (A), σA (h) ⊂ R is equivalent to (e + h2 )−1 ∈ A0 . So, in case a semi-HB∗ -algebra A[τ ] is pseudo-complete, property (2) of Definition 7.5.27 is equivalent to A[τ ] having hermitian involution. Even though, in general, a semi–HB∗ -algebra need not be pseudo-complete, a similar to the aforementioned equivalence implied by Proposition 3.1.5 holds, as Lemma 7.5.30, below, records. Definition 7.5.29 Let A[τ ] be a locally convex algebra, a ∈ A and let (B)A be the family of subsets of A as defined just after Proposition 2.2.8. Then, for B ∈ (B)A , σB (a) denotes the following subset of the extended complex plane C∗ :  σB (a) = λ ∈ C : λe − a has no inverse in A[B]  ∞, if and only if, a ∈ / A[B] . Given the fact that by Proposition 2.2.4, A0 =



A[B], we easily deduce that

B∈(B)A

σA (a) =



σB (a), a ∈ A,

B∈(B)A

where σA (a) is the spectrum of a as defined in Definition 2.3.1. Lemma 7.5.30 Let A[τ ] be a locally convex ∗-algebra which satisfies property (1) of Definition 7.5.27. Then, for every h ∈ H (A), (e + h2 )−1 exists in A0 , if and only if, σB0 (h) ∩ C ⊂ R. Proof ⇒ Let (e + h2 )−1 ∈ A0 , for h ∈ H (A). Then, there is λ > 0 such that n the set C := { λ(e + h2 )−1 : n ∈ N} is bounded with respect to τ . Let B be the  ∗ . Therefore, λ(e + h2 )−1 ∈ B0 , absolutely convex hull of C ∪ {e}. Then, B ∈ B A 2 −1 hence (e + h ) ∈ A[B0 ]. Since,

h(e + h2 )−1

2

−2

−1

−2 = h2 e + h2 = e + h2 − e + h2 ,

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7 Applications I: Miscellanea

2

we have that h(e + h2 )−1 ∈ A[B0 ]. So, the set  

2 n 2  λ−1 h(e+h2 )−1 : n ∈ N is ·B0 -bounded for λ >  h(e+h2 )−1 B . 0

Given that B0 is τ -bounded, 

we have τ ≺  · B0 on A[B0 ] (see  comments after n Definition 2.2.2). Therefore, λ−1 (h(e + h2 )−1 )2 : n ∈ N is τ -bounded, i.e.,

2 h(e + h2 )−1 ∈ A0 . Following the arguments of the proof of Lemma 3.1.4(ii), we then have that h(e + h2 )−1 ∈ A0 . Thus, by the argumentation of the first paragraph of the present proof we have h(e+h2 )−1 ∈ A[B0 ]. Since e+h2 = −(ie+h)(ie−h), it follows that (ie − h) is invertible and

−1 −1 −1 −1





ie − h = − ie + h e + h2 = −h e + h2 − i e + h2 .

−1 Thus, ie − h ∈ A[B0 ], i.e., i ∈ / σB0 (h) (for the latter notation, see Definition 7.5.29). For λ + iμ, λ, μ ∈ R and μ not zero, we have that μ−1(h − λe) is self-adjoint, thus from what is shown previously, i ∈ / σB0 μ−1 (h − λe) . So, λ + iμ ∈ / σB0 (h), from which the result follows. ⇐ Let σB0 (h) ∩ C ⊂ R, for h ∈ H (A). Therefore, i, −i ∈ / σB0 (h), i.e., (ie − h)−1 , (ie + h)−1 ∈ A[B0 ]. Thus, since −(ie + h)(ie − h) = e + h2 , we have that (e + h2 )−1 ∈ A[B0 ] and since A[B0 ] ⊂ A0 , we conclude that (e + h2 )−1 ∈ A0 .   By Theorem 3.3.9(ii), every GB∗ -algebra contains a certain C ∗ -algebra. The following result states that we have the analogous situation in a semi-HB∗ -algebra. Proposition 7.5.31 Let A[τ ] be a semi-HB∗ -algebra. Then A[B0 ] is a C ∗ -algebra with respect to  · B0 . Proof Let h be a self-adjoint element of A[B0 ]. By the hermiticity of involution, (e + h2 )−1 ∈ A0 . Therefore, by following analogous arguments as in the proof of Lemma 7.5.30, (e + h2 )−1 ∈ A[B0 ]. So, A[B0 ] has hermitian involution (see Remark 7.5.28). Then, from the Shirali–Ford theorem (see [141] and/or [60, Theorem 22.23]), A[B0 ] has symmetric involution. The proof from this point onwards follows exactly the proof of Theorem 3.3.9(ii).   Note that in the previous Proposition 7.5.31, continuity of involution is not assumed, since the Shirali–Ford theorem, which is the key result in the proof, is formulated for a Banach algebra with a not necessarily continuous involution. The following result, which is due to Dixon [47, Appendix A, Lemma A. 1], shows that HB∗ -algebras and GB∗ -algebras coincide in the commutative setting. Proposition 7.5.32 (Dixon) If A[τ ] is a commutative HB∗ -algebra, then, A[τ ] is a GB∗ -algebra. Proof Since A is commutative, A[B0 ] = A0 . Let h = h∗ ∈ A. By assumption of

−1

−1 ∈ A[B0 ]. A being an HB∗ -algebra, we have that e + (e − h)2 , e + (e + h)2

7.5 A Vidav-Palmer Theorem for GB*-Algebras

199

−1

−1

So, e + (e − h)2 − e + (e + h)2 ∈ A[B0 ]. Moreover,

−1

−1 − e + (e + h)2 e + (e − h)2 −1

−1



e + (e + h)2 − (e + (e − h)2 ) e + (e + h)2 = e + (e − h)2



−1 −1 = e + (e − h)2 (4h) e + (e + h)2

−1 

= 4h (e + (e + h)2 e + (e − h)2 ) 

−1 1 −1 = 4h 4e + h4 = h e + h4 . 4

Therefore, h(e + 14 h4 )−1 ∈ A[B0 ], for every h ∈ H (A). Hence,

−1 1 √  1 √ 4 −1 h e + h4 =√ 2h e + 2h ∈ A[B0 ], 4 2 that is, h(e + h4 )−1 ∈ A[B0 ], ∀ h ∈ H (A).

(7.5.14)

Let x ∈ A. Then, x can be written as x = h + ik, where h = 12 (x + x ∗ ) and k = 2i1 (x − x ∗ ). Clearly h, k ∈ H (A). Therefore, (e + h4 )−1 , (e + k 4 )−1 ∈ A[B0 ]. Hence, due to (7.5.14) and due to the assumption of commutativity of A, we have that



−1

−1 −1

−1 ∈ A[B0 ] and k1 ≡ k e + h4 ∈ A[B0 ], h1 ≡ h e + h4 e + k4 e + k4 where k1 , h1 are self-adjoint elements in A[B0 ], hence there exists a self-adjoint element l1 ∈ A[B0 ], such that h21 + k12 = l12 . Then,

l1 (e + h4 )(e + k 4 )

2

2

2 = l12 e + h4 e + k 4 

2

2

= h21 + k12 e + h4 e + k 4 = h2 + k 2 .

Therefore, h2 + k 2 = l 2 , where l = l1 (e + h4 )(e + k 4 ) is a self-adjoint element of A. Hence, e + x ∗ x = e + (h − ik)(h + ik) = e + h2 + k 2 = e + l 2 . By assumption, (e + l 2 )−1 ∈ A[B0 ]. So, we conclude that (e + x ∗ x)−1 ∈ A[B0 ], from which it follows that A[τ ] is a GB∗ -algebra.  

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7 Applications I: Miscellanea

A useful “change of perspective” in terms of the topology considered, enables the passage from a semi-HB∗ -algebra to an HB∗ -algebra, as described in the following Proposition 7.5.33 Let A[τ ] be a semi-HB∗ -algebra. Then, there is a topology τ

on A, such that A[τ ] is an HB∗ -algebra and moreover the greatest members of the

∗ ∗ respective collections B A[τ ] and BA[τ ] coincide; i.e., in symbols B0 (τ ) = B0 (τ ) (in this regard, see also Corollary 6.3.7). Proof Let U be a base of absolutely convex neighbourhoods of 0 for the topology τ . Then, by considering NU = U ∩ U ∗ for each U ∈ U, we get a base of neighbourhoods of 0 for a locally convex topology, τ say, of A. Then, it is easily seen that τ ≺ τ , the involution on A is τ -continuous, B0 is τ -closed and B0 is  ∗ . Therefore, since the completeness of A[B0 ], and the the greatest element of B A[τ ] fact that (e + h2 )−1 ∈ A0 , for every h ∈ H (A), remain valid under this change of topology (for the latter argument see Lemma 7.5.30), we conclude that A is an HB∗ -algebra with respect to τ .    The next result can be seen as “the half way” for a Vidav–Palmer type theorem for a semi-HB∗ -algebra. Theorem 7.5.34 Let A[τ ] be a hypocontinuous semi-HB∗ -algebra. Then, there exists a defining family of seminorms 

for τ , equivalent to the given one , such that for every self-adjoint element h in A, W

(h) ⊂ R. Proof We first show that there is a defining family of seminorms 

for τ , equivalent to the initial one , such that xB0 = x , for every normal element x in

 ∗ . From Lemma 7.5.10, A[B0 ], where B0 corresponds to the largest element of B A there exists such a defining family of seminorms 

for τ , such that B0 ⊂ C

. Consequently, A[B0 ] ⊂ A[C

] and xB0 ≥ x

, for every x ∈ A[B0 ]. From Proposition 7.5.31, A[B0 ] is a C ∗ -algebra. Hence, it follows from [127, Theorem (4.8.3)] that r

(x) = rB0 (x), for every x ∈ A[B0 ], where r

(x), rB0 (x) denote the spectral radii of x in A[C

], A[B0 ], respectively. If, in addition, x is a normal element in A[B0 ], then xB0 = rB0 (x), consequently xB0 = r

(x) ≤ x

. Therefore, we conclude that xB0 = x

, for every normal element x in A[B0 ]. Let us consider now a self-adjoint element h in A[B0 ] and let t ∈ R. Then, e+ith is a normal element in A[B0 ]. Therefore, by the argument of the previous paragraph ∗ we have that e + ith

= e + ithB0 . Since  · B0 is a C -norm on A[B0 ] we have 2 2     e + ith = e + ithB = (e − ith)(e + ith)B

0 0     2 2 2 2   = e+t h B = e+t h  .

0

7.5 A Vidav-Palmer Theorem for GB*-Algebras

201

Therefore,     1 1  e + ith − 1 = lim 1 e + t 2 h2  2 − 1 = 0. 



t →0 t t →0 t lim

Thus, from [30, Lemma 5.2] it follows that V (A[C

], h) ⊂ R. Hence, by Proposition 7.5.8, we have that W (h) ⊂ R.

From Proposition 7.5.33, there is a topology τ finer than the topology τ and such that A[τ ] is an HB∗ -algebra. Let h be a self-adjoint element in A and let Ch be a maximal commutative ∗-subalgebra of A[τ ], which contains h. Then, by Proposition 7.5.32, Ch [τ ] is a GB∗ -algebra. From the proof of Theorem 4.2.11, there is a sequence {hn }n of self-adjoint elements in A[B0 ], such that hn → h. τ

Hence, hn → h. By the argument of the previous paragraph, W

(hn ) ⊂ R. So, τ

from Proposition 7.5.6, we conclude that W

(h) ⊂ R.

 

We shift now our attention to the inverse direction, that is, we look at algebras that are spanned as linear spaces by their hermitian elements and investigate whether this property can characterize the nature of the algebras. To begin with, we state the following helpful Lemma 7.5.35 Let (A, ) be a complete locally convex algebra with the property that every a ∈ A is written as a = h + ik, where h, k ∈ H. Then, the elements h, k in the decomposition of a are uniquely determined. Proof Let h, k ∈ H, such that h + ik = 0. Then, it is straightforward to check that W (h) = −iW (k). Hence, W (h), W (k) ⊂ R ∩ iR = {0}, therefore W (h) = {0} = W (k). So, by Corollary 7.5.20, we have that h = k = 0.   Based on the previous lemma, it is meaningful to write A = H + iH, in case (A, ) is a complete locally convex algebra, which is spanned, as a linear space, by its hermitian elements. In this case, we say that (A, ) has a hermitian decomposition. The next step we are going to take is to show that A possesses an involution. Proposition 7.5.36 Let (A, ) be a complete locally convex algebra with a hermitian decomposition. Then, there is a map  : A → A, such that (i) (a  ) = a, a ∈ A; (ii) (a + b) = a  + b  , a, b ∈ A; (iii) (λa) = λa  , λ ∈ C, a ∈ A. Proof Let a ∈ A. Then, there are unique hermitian elements h, k ∈ H, such that a = h + ik. Consider the map  : A → A : a  = h − ik. It is straightforward to check that the map  enjoys the desired properties.   While the establishment of a vector space involution on A is straightforward, as shown in Proposition 7.5.36, the proof that this involution is also algebraic (i.e.,

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7 Applications I: Miscellanea

(ab) = b a  , a, b ∈ A) requires some more effort. Towards this direction, we first have the following result. Proposition 7.5.37 Let (A, ) be a complete locally convex algebra with a hermitian decomposition. Then, A[C ] is a C ∗ -algebra with unit ball C . Proof Let a ∈ A[C ]. Since A has a hermitian decomposition, there are unique elements h, k ∈ H, such that a = h + ik. We show that h, k ∈ A[C ]. The latter fact will follow, once we show that W (h), W (k) are bounded (for this, see the proof of Theorem 7.5.14). By Proposition 7.5.8,

 

  W (a) ⊂ V A[C ], a ; moreover V A[C ], a ⊂ λ : |λ| ≤ a , where the second inclusion in the previous relation is due to the following facts: by Proposition 7.5.8, W (a) ⊂ V (A[C ], a). The element λ ∈ V (A[C ], a) can be written as λ = f (a), for some f ∈ A[C ] with f  = f (e) = 1 (see the paragraph preceding Proposition 7.5.8). Hence, |λ| = |f (a)| ≤ f a = a . Therefore, W (a) ⊂ {λ ∈ C : |λ| ≤ a }.

 Now from the inclusion W (a) ⊂ V A[CΓ ], a as above, W (a) is bounded. Since, as can be easily checked, W (h) = Re(W (a)) and W (k) = Im(W (a)), so we have that W (h), W (k) are bounded. Thus, h, k ∈ A[C ]. From Proposition 7.5.19, 



V A[C ], h ⊂ coW (h) and V A[C ], k ⊂ coW (k). So, V (A[C ], h), V (A[C ], k) ⊂ R. Therefore, the normed algebra A[C ] satisfies the conditions of the Vidav–Palmer theorem (see Theorem 7.5.1), from which follows that A[C ] is a C ∗ -algebra with unit ball C .   In the course of proving that the involution of Proposition 7.5.36 is algebraic, it would be useful to show that for certain elements of a hypocontinuous complete locally convex algebra, the closed convex hulls of the sets of the spectrum and of the numerical range coincide, a result which clearly has its independent interest (see Proposition 7.5.39). The following rather technical result is needed for the proof of Proposition 7.5.39. The proof of Proposition 7.5.38 is based on a mere recollection of results from Chaps 1 and 2. We recall that (B)A denotes the family of subsets of A as described in the paragraph following Proposition 2.2.8.

7.5 A Vidav-Palmer Theorem for GB*-Algebras

203

Proposition 7.5.38 Let A[τ ] be a commutative locally convex algebra and B ∈ (B)A , such that A[B] is a C ∗ -algebra. Let a = h + ik ∈ A (h, k ∈ H (A)) be such that   σB (a) ∩ C ⊂ z : Rez ≥ 0 , σB (h) ∩ C ⊂ R and σB (k) ∩ C ⊂ R. Then, σB (h) ∩ C ⊂ [0, ∞). Proof Let M be the carrier space of A[B] and let  = {x ∈ A : σB (x) = C∗ }. By using the arguments of the proof of Proposition 2.5.4 we can easily see that each ϕ ∈ M can be extended to a C∗ -valued function ϕ on , such that ϕ has the properties (1)–(4) of Proposition 2.5.4. Moreover, the proof of Proposition 2.5.5 carries over with exactly the same arguments, for A[B] in place of A0 . Therefore, σB (a) = {ϕ (a) : ϕ ∈ M}. By assumption,

−1

−1 ∈ A[B]. a = h + ik and σB (h) ∩ C ⊂ R, so ie − h , ie + h Thus, −1 −1

−1

ie + h = − ie − h ∈ A[B]. e + h2 An analogous argument for k shows that (e + k 2 )−1 ∈ A[B]. Now, since (e + (e + k 2 )−1 ∈ A[B], imitating the proof of Lemma 3.4.6, we conclude that the sets

h2 )−1 ,

    ϕ ∈ M : ϕ(h) = ∞ , ϕ ∈ M : ϕ(k) = ∞ are closed, nowhere dense subsets of M (note that the assumption of A[B] being a C ∗ -algebra is crucial in this point). Hence, by considering ϕ in the dense subset of M, where both ϕ (h), ϕ (k) are finite, we have that ϕ (a) = ϕ (h) + iϕ (k). So, since ϕ (h) ∈ σB (h) ∩ C ⊂ R and ϕ (k) ∈ σB (k) ∩ C ⊂ R, we conclude that ϕ (h) = Re(ϕ (a)), which is not negative by assumption. Thus, σB (h) ∩ C ⊂ [0, ∞).   Proposition 7.5.39 Let (A, ) be a hypocontinuous complete locally convex algebra with a hermitian decomposition. Let a ∈ A be  an element, such that a = h + ik, h, k ∈ H and hk = kh. Then, co σ (a) ∩ C = co W (a). Proof By Theorem 7.5.14, σ (a) ∩ C ⊂ W (a). The proof can be completed by an implication of the Hahn–Banach separation theorem, once it is shown that the relation σ (a) ∩ C ⊂ {z : Rez ≥ 0} implies that W (a) ⊂ {z : Rez ≥ 0}. Given the fact that h, k commute, we can suppose without loss of generality that

204

7 Applications I: Miscellanea

the algebra A is commutative. Moreover, by Proposition 7.5.37, A[C ] is a C ∗ algebra. We therefore note that all the conditions of Proposition 7.5.38 are satisfied, hence σ (h) ∩ C ⊂ [0, ∞). By Theorem 7.5.26, we then have that W (h) ⊂ [0, ∞). Since k ∈ H and thus W (ik) ⊂ iR, we conclude that W (a) ⊂ {z : Rez ≥ 0}.   Theorem 7.5.40 Let (A, ) be a hypocontinuous complete locally convex algebra with a hermitian decomposition. Then, the map  of Proposition 7.5.36 is an algebraic involution. Proof We first show that for h ∈ H, h2 ∈ H. Indeed, for h2 , there are unique u, v ∈ H, such that h2 = u + iv. Then, h(u + iv) = (u + iv)h, so that hu − uh = i(vh − hv). By Theorem 7.5.24, hu − uh, i(hu − uh) ∈ H, so i(vh − hv), vh − hv ∈ H. Therefore, W (i(vh − hv)) ⊂ R ∩ iR = {0}, which from Corollary 7.5.20 implies that i(hv − vh) = 0. Then, hu = uh so that h2 u = h(uh) = (hu)h = uh2 . Hence, (u+ iv)u = u(u + iv),

 if and only if, uv = vu. By Proposition 7.5.39, co W (h2 ) = co σ (h2 ) ∩ C . Hence,



2

2 co W (h2 ) = co (σ (h) ∩ C) ) ⊂ co W (h) ⊂ R. The last but one inclusion in the aforementioned string of relations is due to Theorem 7.5.14. Therefore, we conclude that W (h2 ) ⊂ R; hence h2 ∈ H. Now let a, b ∈ A. Based on the relation ab = 12 (ab +ba  )−i 12 i(ab −ba  ), we show that ab + b a  ∈ H. Indeed, let a = h + ik, b = u + iv for unique hermitian elements h, k, v, v. Then, ab + b  a  = (hu + uh) − (kv + vk) + i(ku − uk) + i(hv − vh). By Theorem 7.5.24, i(ku − uk), i(hv − vh) ∈ H. Moreover, by the argument of the previous paragraph, k 2 , v 2 , (k + v)2 , u2 , h2 , (h + u)2 ∈ H, hence kv + vk, hu + uh ∈ H. Therefore, ab + b a  ∈ H. Similarly, it can be shown that i(ab − b a  ) ∈ H. Thus, based on the definition of the map , we deduce that  

 1  1 ab + b  a  + i i ab − b  a  = b a  ab = 2 2  

7.5 A Vidav-Palmer Theorem for GB*-Algebras

205

Our next objective is to show that for a hypocontinuous complete locally convex  ∗ attains a greatest member. algebra with hermitian decomposition, the family B A The following two results are devoted to this goal. Theorem 7.5.41 Let (A, ) be a commutative hypocontinuous complete locally convex algebra with a hermitian decomposition and let B ∈ (B)A . Then, B ⊂ C . Proof Since A is commutative and complete, the family (B)A is outer-directed with respect to inclusion (see Theorem 2.2.10). Hence, there is a set F ∈ (B)A , such that B ∪ C ⊂ F . By Lemma 7.5.10, there is a defining family of seminorms  for the topology τ corresponding to the initial family on A, such that F ⊂ C . It suffices to show that C ⊂ C . By C ⊂ C , we have that A[C ] ⊂ A[C ]. Therefore, if a ∈ A and λ ∈ / σ (a) ∩ C, then (λe − a)−1 ∈ A[C ] and so (λe − a)−1 ∈ A[C ]. Thus, σ (a) ∩ C ⊂ σ (a) ∩ C. Since A is commutative and has a hermitian decomposition, Proposition 7.5.39 yields that co(W (a)) ⊂ co(W (a)). The previous inclusion gives us, in particular, that H ⊂ H . Let a ∈ A[C ]. Then, a = h + ik, where h, k ∈ H and thus a fortiori h, k ∈ H . By the second paragraph of the proof of Proposition 7.5.19, we have that W (a) ⊂ {λ : |λ| ≤ a }. So, W (a) is bounded, hence W (h), W (k) are bounded. Thus, h, k ∈ A[C ]. So, a  = h − ik ∈ A[C ] and thus A[C ] is a Banach algebra with involution . If h is a self-adjoint element of A[C ], then h ∈ H . Indeed, if h = u + iv, where u, v ∈ H then, by h = h , we have that v = 0, hence h = u ∈ H . Now, from Theorem 7.5.22 and Corollary 7.5.23, we have that  exp(ih) ≤  exp(ih) = 1. Since  exp(ih) ≥  exp(ih)  exp(−ih) ≥  exp(ih) exp(−ih) = 1, we obtain that  exp(ih) = 1. Then, from [126, Theorem 10.1] follows that A[C ] is a C ∗ -algebra with unit ball C . According to [126, Theorem 9.7], C  is the closed convex hull, with respect to  ·  , of the set L = exp(ia) : a ∈   H A[C ] . Clearly exp(ih) ∈ L and since  exp(ih) = 1,

 exp(ih) ∈ U A[C ] ⇒ exp(ih) ∈ C ⇒ L ⊂ C . By the way it is defined, C is closed with respect to the topology τ of A and C ⊂ C by our initial assumption. Since τ ≺  ·  on A[C ], C is  ·  closed. Therefore, by L ⊂ C , it follows that C ⊂ C . Therefore, B ⊂ C ⊂ C .   Theorem 7.5.42 Let (A, ) be a hypocontinuous complete locally convex algebra  ∗ be arbitrary. Then, B ⊂ C ; i.e., with a hermitian decomposition and let B ∈ B A ∗ . C is a greatest member in B A

206

7 Applications I: Miscellanea

Proof Let h ∈ H ∩ B. Then B, the closure of B in (A, ), belongs to (B)A . Since h ∈ H, h is a self-adjoint element of A with respect to the involution . If we consider a maximal commutative -subalgebra of A containing h, then by Theorem 7.5.41, we have that h ∈ C . Now, let x = h + ik ∈ B, where h, k ∈ H. Since x  ∈ B  = B, h − ik ∈ B  (B denotes the set of all x  for x ∈ B). Since B is convex, 12 x  + 12 x ∈ B, from which follows that h ∈ B. Similarly, since B is absorbent, 1i x, 1i x  ∈ B, thus 1 1 1 1  2 ( i x) + 2 (− i x ) = k ∈ B. So, h, k ∈ H ∩ B. By the argument of the previous paragraph, we then have that x ∈ 2C . Therefore, B is a bounded subset of A[C ] with respect to  ·  , such that B 2 ⊂ B and B  = B. By Proposition 7.5.37, A[C ] is a C ∗ -algebra with unit ball C . Hence, C is the largest element, which is  bounded, self-adjoint (i.e., C = C ) and submultiplicative (i.e., C 2 ⊂ C ). This can be easily seen as follows: for x ∈ B, x  x ∈ B  B = B 2 ⊂ B and by induction (x  x)n ∈ B, n ∈ N. Suppose that x ∈ / C . Then,  !      n    x  > 1 and so x  x ! = x 2 > 1; therefore (x  x)2n  = x  x 2 → ∞,





n→∞

n

which is a contradiction given that (x  x)2 ∈ B and B is a bounded subset of A[C ].   The following Theorem 7.5.43 is a Vidav-Palmer type theorem for semi-HB∗ algebras. Theorem 7.5.43 (Wood) Let A[τ ] be a hypocontinuous complete locally convex algebra. Then, A[τ ] is a semi-HB∗ -algebra, if and only if, there is a defining family of seminorms for τ , such that (A, ) has a hermitian decomposition. Proof ⇒ By Theorem 7.5.34, there is a defining family of seminorms 

for τ , (h) ⊂ R, for every h ∈ H (A). It equivalent to the initial family , such that W

follows then that H (A) ⊂ H . Hence, since every a ∈ A can be expressed as a





linear combination of elements in H (A), namely a = 12 a + a ∗ + i 2i1 a − a ∗ ,  we have that A = H

+ iH

. The uniqueness of the latter decomposition of (A, ) follows from Lemma 7.5.35. ⇐ Let be a defining family of seminorms for τ , with respect to which (A, ) has a hermitian decomposition. Then, by Theorems 7.5.40, 7.5.42 and Proposition 7.5.37, A has an involution , C is the greatest member of the family  ∗ , with respect to inclusion, and A[C ] is a C ∗ -algebra. Let h ∈ H . Then, since B A h clearly satisfies the condition of Proposition 7.5.39, we have that σ (h) ∩ C ⊂ co(σ (h) ∩ C) = co(W (h)) ⊂ R. Therefore, by Lemma 7.5.30, the involution  is such that (e + h2 )−1 ∈ A0 , for every h in A, with h = h . Thus, we conclude that A[τ ] is a semi-HB∗ -algebra.  

7.5 A Vidav-Palmer Theorem for GB*-Algebras

207

Corollary 7.5.44 Let A[τ ] be a hypocontinuous complete semi-HB∗ -algebra. Then, there is a defining family of seminorms 

for τ , equivalent to the given , with respect to which H = H (A).

Proof By the first part of the proof of Theorem 7.5.43 we have that there is a defining family of seminorms 

, equivalent to the given , such that H (A) ⊂ H

. On the other hand, let h ∈ H . Since

h=

   1  1 1 1 h + h∗ + i h − h∗ and h + h∗ , h − h∗ ∈ H (A) ⊂ H

, 2 2i 2 2i

we have that    1 1 ∗ h − ∈ R ∩ iR. h + h∗ ∈ H h + h ∩ iH , hence W  



2 2

 So, by Corollary 7.5.20, h = 12 h + h∗ equivalently h = h∗ , i.e., h ∈ H (A). h−

 

Theorem 7.5.43 can be further enhanced. Indeed, the next three lemmas culminate in Proposition 7.5.48, which shows that every complete hypocontinuous semi-HB∗ algebra is actually a semi-GB∗ -algebra. We note that, in accordance with the notation introduced in Chap. 3 (see comments after Definition 3.1.2), a self-adjoint element h of a semi-HB∗ -algebra A is called positive, denoted by h ≥ 0, if σA (h) ∩ C ⊂ [0, ∞). Lemma 7.5.45 Let A[τ ] be a hypocontinuous, complete semi-HB∗ -algebra and let h, k ∈ A with h, k ≥ 0. Then, h + k ≥ 0. Proof By Corollary 7.5.44, there is a defining family of seminorms 

for τ , such that H = H (A). The assumption of positivity for h, k and Theorem 7.5.26, lead us

to the fact that W (h) and W (k) are contained in [0, ∞). So, W (h + k) ⊂ [0, ∞).  



Therefore, with the aid of Proposition 7.5.39, we have that σA (h + k) ∩ C = σ

(h + k) ∩ C ⊂ co(σ

(h + k) ∩ C) = coW

(h + k) ⊂ R. Hence, we conclude that h + k ≥ 0.

 

In the proof of the following result, as far as quasi-invertible elements are concerned, the reader is referred to the second last paragraph before Definition 2.1.1. Lemma 7.5.46 Let A[τ ] be a locally convex algebra with identity and let x, y ∈ A. If λ ∈ C∗ (λ = 0). Then λ ∈ σA (xy), if and only if, λ ∈ σA (yx). Proof We first show that β(xy) = β(yx): let λ > 0, such that (λ−1 xy)n → 0, for n → ∞. By induction, (λ−1 xy)n = λ−n x(yx)n−1y, n ∈ N. So, by continuity of multiplication we have that λ−n (yx)n+1 → 0, for n → ∞. Thus,

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7 Applications I: Miscellanea

by Proposition 2.2.14 (3), we have that β(xy) ≥ β(yx). The reverse inequality is derived by symmetrical arguments. Let ∞ ∈ σA (xy). That is, xy ∈ / A0 . By Proposition 2.2.14 (2) we equivalently have that β(xy) = ∞. By the previously established equality, β(yx) = ∞. Thus, yx ∈ / A0 , hence ∞ ∈ σA (yx). Let λ ∈ C\{0}, such that λ ∈ / σA (xy). Then, (λe −xy)−1 ∈ A0 . Therefore, since −1 e − λ xy is invertible, we have that λ−1 xy is quasi-invertible, with quasi-inverse (λ−1 xy)◦ = e − (e − λ−1 xy)−1 . By the following equality, 



λ−1 yx ◦ λ−1 y(λ−1 xy)◦ x − λ−1 yx = λ−1 y λ−1 xy ◦ (λ−1 xy)◦ x, we then have that λ−1 yx is quasi-invertible, with quasi-inverse (λ−1 yx)◦ = λ−1 y(λ−1 xy)◦ x − λ−1 yx. Hence, e − λ−1 yx is invertible with inverse (e − λ−1 yx)−1 = e − (λ−1 yx)◦ = e + λ−1 y(e − λ−1 xy)−1 x. Furthermore, we have 



 β λ−1 y(e − λ−1 xy)−1 x = β λ−1 xy(e − λ−1 xy)−1 

= β (e − λ−1 xy − e)(e − λ−1 xy)−1 

= β e − (e − λ−1 xy)−1 

≤ β(e) + β (e − λ−1 xy)−1 < ∞, where the one to last inequality is due to Proposition 2.2.15. Therefore, λ−1 y(e − λ−1 xy)−1 x ∈ A0 . We conclude that (e − λ−1 yx)−1 ∈ A0 , i.e., λ ∈ / σA (yx). By exchanging the roles of x and y, we have that for λ ∈ C \ {0}, λ ∈ σA (xy), if and only if, λ ∈ σA (yx).   Lemma 7.5.47 Let A[τ ] be a complete hypocontinuous semi-HB∗ -algebra, and x ∈ A. If xx ∗ ≤ 0, then xx ∗ = 0. Proof We consider a defining family of seminorms for τ , such that H = H (A), as is provided by Corollary 7.5.44. Let x = h + ik, where h, k ∈ H (A). Then, xx ∗ = 2(h2 + k 2 ) − x ∗ x.

(7.5.15)

By using similar arguments to the proof of Proposition 2.5.5, we can see that σA (h2 ) ∩ C and σA (k 2 ) ∩ C are contained in [0, ∞). Hence, h2 , k 2 ≥ 0. Moreover, by Lemma 7.5.46 and from the assumption ‘xx ∗ ≤ 0’, we have that −x ∗ x ≥ 0. Therefore, Lemma 7.5.45 and (7.5.15) imply xx ∗ ≥ 0. So, we have that xx ∗ ≤ 0 and

7.5 A Vidav-Palmer Theorem for GB*-Algebras

209

xx ∗ ≥ 0, from which it follows that σA (xx ∗ ) ⊂ {0, ∞}. By considering a maximal commutative ∗-subalgebra of A that contains xx ∗ , we may suppose without loss of generality that A is commutative. By Theorem 3.4.9 and Lemma 3.4.6, the Gelfand transform, xx ∗ , of xx ∗ is a continuous C∗ -valued function on the carrier space of A0 = A[B0 ], that has as possible values the points 0 and ∞; moreover, it takes the value ∞ on a nowhere dense subset. Therefore, we conclude that xx ∗ = 0, hence xx ∗ = 0.   Proposition 7.5.48 Let A[τ ] be a complete hypocontinuous semi-HB∗ -algebra. Then, A[τ ] is a semi-GB∗ -algebra. Proof Let x ∈ A. The element x ∗ x is contained in a commutative semi-HB∗ algebra, say C. By Proposition 7.5.32, C is a semi-GB∗ -algebra. Hence, from Theorem 4.1.4(ii), there exist h, k ≥ 0, such that x ∗ x = h − k and hk = kh = 0. Since (xk)∗ (xk) = k(x ∗ x)k = −k 3 ≤ 0, Lemma 7.5.47 gives that (xk)∗ (xk) = 0. Thus, k = 0 and so x ∗ x = h ≥ 0. Thus, A is symmetric (see the discussion before Definition 3.1.3), i.e., A is a semi-GB∗ -algebra.   Based on the aforementioned result, Theorem 7.5.43 can take the following improved form. In particular, Theorem 7.5.49 is a Vidav-Palmer type theorem for semi-GB∗ -algebras. Theorem 7.5.49 (Wood) Let A[τ ] be a complete hypocontinuous locally convex algebra. Then, A[τ ] is a semi-GB∗ -algebra, if and only if, there is a defining family of seminorms for τ , with respect to which (A, ) has a hermitian decomposition. Corollary 7.5.50 Every hypocontinuous complete GB∗ -algebra has a defining family of seminorms, with respect to which the algebra has a hermitian decomposition. In the case where A[τ ] is metrizable, we can deduce a more general “if and only if” argument than that of Theorem 7.5.49, as the next result informs us, which namely is a Vidav–Palmer type theorem for GB∗ -algebras. Theorem 7.5.51 (Wood) Let A[τ ] be a Fréchet locally convex algebra. Then, A[τ ] is a GB∗ -algebra, if and only if, there exists a defining family of seminorms with respect to which (A, ) has a hermitian decomposition. Proof ⇒ Since A[τ ] is Fréchet, the multiplication of A[τ ] is jointly continuous (see comments following Definition 3.1.1). This results in A[τ ] being hypocontinuous. Indeed, let U be a 0-neighbourhood and B a bounded subset in A[τ ]. By joint continuity of multiplication, there is a 0-neighbourhood V , such that V V ⊂ U . Since B is bounded, there is ε > 0, such that B ⊂ εV . Hence, for the 0neighbourhood 1ε V , we have that B

1  1  1  V ⊂ εV V = V V ⊂ U and similarly V B ⊂ U, ε ε ε

i.e., A[τ ] is hypocontinuous. Then, the proof of this direction follows immediately from Theorem 7.5.49.

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7 Applications I: Miscellanea

⇐ suppose that there exists a defining family of seminorms for τ , with respect to which (A, ) has a hermitian decomposition. By Theorems 7.5.43 and 7.5.49, A[τ ] is a semi-GB∗ -algebra with respect to the involution . Thus, what remains to be shown in order for A[τ ] to be a GB∗ -algebra is the continuity of the involution . We first note that by the definition of  it is straightforward that H (A) = H . Since A[τ ] is Fréchet and  is conjugate linear we can apply the closed graph theorem [74, p. 301, Theorem 4]. Therefore, the continuity of the map  will follow once it has a  closed graph. So, let (xn ) be a sequence in A[τ ], such that xn → x and xn → y, where x, y ∈ A. Then,

 x + y = lim xn + xn ∈ H (A) = H = H = H (A), n

where the next to last equality follows from Proposition 7.5.6. Moreover, 



i x − y = lim i xn − xn ∈ H (A) = H = H = H (A). n

Therefore, we have that



 (x + y) = x + y and i x − y = i(x − y) , from which we obtain y = x  . This shows that  has a closed graph and so the proof is complete.   We remark that for the left-hand side direction (⇐) in the proof of the previous Theorem 7.5.51, the hermitian decomposition together with the metrizability of the algebra guarantee the continuity of the involution , a property which is encoded in the “DNA” of a GB∗ -algebra. Notes The results of this section are due to A.W. Wood and can be found in [156]. Proposition 7.5.32 is due to P.G. Dixon and comes from [47, Appendix A, Lemma A.1]. We remark that, the concept of a GB∗ -algebra, defined by A.W. Wood (Definition 7.5.27), for the needs of the Vidav–Palmer Theorem in the context of GB∗ -algebras (see, for example, Theorem 7.5.51), is equivalent to that of a GB∗ algebra of P.G. Dixon as given by Definition 3.3.5.

Chapter 8

Applications II: Tensor Products

Tensor products of GB∗ -algebras were investigated for the first time in [64]. As we have noticed, GB∗ -algebras being algebras of unbounded operators are, in particular, important for mathematical physics and quantum mechanics. In this aspect, we want to emphasize that there is a physical justification for using tensor products. For example, tensor products are used to describe two quantum systems as one joint system (see [2]), while the physical significance of tensor products always depends on the applications, which may involve wave functions, spin states, oscillators etc.; see e.g., [29, 70]. Other results on topological tensor products of unbounded operator algebras can be found in [1, 65, 71].

8.1 Prerequisites: Examples  We emphasize that throughout this chapter we deal with locally convex (∗-)algebras with jointly continuous multiplication.

Let us now recall some standard definitions concerning topological tensor products. Definition 8.1.1 ([60, Definition 29.5]) Let A1 [τ1 ], A2 [τ2 ] be locally convex ∗algebras. Let A1 ⊗ A2 be their (algebraic) tensor product ∗-algebra. A topology τ on A1 ⊗ A2 is called ∗-admissible (with the tensor product ∗-algebra structure of A1 ⊗ A2 ), if the following conditions are satisfied: (1) A1 ⊗ A2 endowed with τ is a locally convex ∗-algebra, with jointly continuous A2 ,. multiplication, denoted by A1 ⊗ A2 and its completion by A1 ⊗ τ

τ

(2) The tensor map ⊗ : A1 × A2 → A1 ⊗ A2 : (x, y) → x ⊗ y is continuous. τ

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Fragoulopoulou et al., Generalized B*-Algebras and Applications, Lecture Notes in Mathematics 2298, https://doi.org/10.1007/978-3-030-96433-7_8

211

212

8 Applications II: Tensor Products

(3) For any equicontinuous subsets M, N of the duals A∗1 , A∗2 of A1 , A2 respectively, the set M ⊗ N = {f ⊗ g : f ∈ M, g ∈ N} is an equicontinuous subset of the dual of A1 ⊗ A2 . τ

When A1 , A2 have no involution, then τ is called a compatible topology on A1 ⊗ A2 , see [111, p. 375, Definition 3.1]. Namely, a tensor topology τ on A1 ⊗ A2 is ∗admissible (resp. compatible), if and only if, ε ≺ τ ≺ π

(8.1.1)

(see [60, p. 370, discussion before 29.7]), where ε, π stand for the biprojective (equivalently injective, cf. also [111]), respectively projective locally convex tensor topologies. Suppose that A1 [τ1 ], A2 [τ2 ] are locally convex (∗-)algebras, with corresponding defining families of (∗-)seminorms, given by 1 = {p}, respectively by 2 = {q}. Then, the defining families of (∗-)seminorms for ε, π, denoted by

⊗ε , respectively ⊗π are given, as ! "!  # ! n ! f (xi )g(yi )!! : f ∈ Up◦ , g ∈ Uq◦ , εp,q (z) := sup !! i=1

z=

n 

xi ⊗ yi ∈ A1 ⊗ A2 , ε

n=1

πp,q (z) := inf

" n n=1

p(xi )q(yi ) : z =

n  n=1

# xi ⊗ yi , z ∈ A1 ⊗ A2 , π

for all (p, q) ∈ 1 × 2 , where infimum (in the last equality) is taken over all representations nn=1 xi ⊗ yi of z in A1 ⊗ A2 . Note that Uk◦ , k = p, q, is the polar π   of the closed unit semiball Uk := xi ∈ Ai : k(xi ) ≤ 1 , k = p, q in Ai , i = 1, 2. In particular, one has that εp,q (x ⊗ y) = p(x)q(y) = πp,q (x ⊗ y), ∀ x ∈ A1 , y ∈ A2 . Clearly, ε, π are Hausdorff, compatible (resp. ∗-admissible) tensor topologies on A1 ⊗ A2 . In the case when Ai [τi ], i = 1, 2 are normed (∗-)algebras, the tensor topologies ε, π are respectively denoted by λ and γ and the corresponding (∗-)norms by  · λ and  · γ . Coming back to Ai [τi ], i = 1, 2, as C ∗ -convex algebras, observe that the injective, respectively projective C∗ -convex tensor topologies α and ω on A1 ⊗ A2 [60, Section 31] are analogous to the minimal (resp. maximal) C ∗ -crossnorms max and min on the tensor product of two C ∗ -algebras [144, p. 203]. Both are Hausdorff ∗-admissible tensor topologies. If A1 [τ1 ], A2 [τ2 ] are pro-C ∗-algebras, let 1 = {p} and 2 = {q} be defining families of C ∗ -seminorms for τ1 , τ2 respectively. Denote

8.1 Prerequisites: Examples

213

by ⊗α = {αp,q }, ⊗ω = {ωp,q } the families of C ∗ -seminorms defining the C ∗ -convex tensor topologies α and ω, respectively. Particularly, let Rk (Ai ) and A2 ), k = p, q, i = 1, 2, stand for the sets of continuous Hilbert space Rp,q (A1 ⊗ π

A2 , whose continuity representations μi on Ai [τi ], i = 1, 2 and μ1 ⊗ μ2 on A1 ⊗ π

is respectively given by the inequalities μ1 (x) ≤ p(x), μ2 (y) ≤ q(y) and (μ1 ⊗ μ2 )(z) ≤ πp,q (z), for all x ∈ A1 , y ∈ A2 and z ∈ A1 ⊗ A2 . Then, π



 A2 αp,q (z) := sup (μ1 ⊗ μ2 )(z) : (μ1 , μ2 ) ∈ Rp (A1 ) × Rq (A2 ) , ∀ z ∈ A1 ⊗ 

π



A2 ) , ∀ z ∈ A1 ⊗ A2 . and ωp,q (z) := sup μ(z) : μ ∈ Rp,q (A1 ⊗ π

π

In this respect, we have that (see [60, Proposition 31.3]) ε ≺ α ≺ ω ≺ π and

(8.1.2)

αp,q (x⊗y) = p(x)q(y) = ωp,q (x⊗y), for every elementary tensor x⊗y in A1 ⊗A2 and any p, q. We proceed now with some examples of topological tensor product ∗-algebras that are, in particular, GB∗ -algebras. Example 8.1.2 Let A1 [τ1 ], A2 [τ2 ] be two pro-C ∗ -algebras (hence GB∗ -algebras). Consider the C ∗ -convex algebras A1 ⊗ A2 , A1 ⊗ A2 . Then, their completions, α

ω

A2 , A1 ⊗ A2 are pro-C ∗ -algebras, therefore GB∗ denoted respectively by A1 ⊗ α

ω

algebras too [60, p. 386]. Example 3.3.16(6) also serves as a topological tensor product that, in particular, is a GB∗ algebra. In order to proceed to our third example we need some prerequisites, which we first exhibit. Let X be a locally compact space and A[τ ] a complete GB∗ algebra with a natural family {pν }ν∈ of ∗-seminorms (see Remark 3.5.7(4) and take, for instance, A[τ ] to be a C ∗ -like locally convex ∗-algebra; in this respect, cf. also Corollary 3.5.4 and Proposition 3.5.8). Consider the locally convex ∗algebra C(X, A) of all A-valued continuous functions on X, with the compact-open topology. According to our conventions, A[τ ] as a locally convex ∗-algebra has a jointly continuous multiplication, therefore so does C(X, A) too. It is known that  A, C(X, A) = Cc (X)⊗ ε

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8 Applications II: Tensor Products

up to an isomorphism of locally convex ∗-algebras [111, p. 391, Theorem 1.1], where Cc (X) is the pro-C ∗ -algebra of all complex-valued continuous functions on X, under the topology “c” of uniform convergence on the compacta of X. As we have already seen, Cc (X) as a pro-C ∗-algebra is a GB∗ -algebra. If K is the family of all compact subsets of X, the topology “c” is determined by the C ∗ -seminorms   qK (f ) = sup |f (x)| : x ∈ K , f ∈ Cc (X), ∀ K ∈ K. Suppose now that τ is defined by a directed family, (pν )ν∈ , of ∗-seminorms; then the functions   qK,ν (f ) = sup pν (f (x)) : x ∈ K , f ∈ C(X, A), K ∈ K, ν ∈ , are ∗-seminorms determining the compact-open topology on C(X, A). With X as before, let Cb (X) denote the C ∗ -algebra of all bounded continuous complex-valued functions on X. Lemma 8.1.3 Let X be a locally compact space and A[τ ] a GB∗ -algebra with a natural family {pν }ν∈ of ∗-seminorms. Let C(X, A), Cb (X) be as above, and A0 =  A[B0 ], where λ is the normed injective tensor topology on Cb (X) ⊗ A[B0 ]. Cb (X)⊗ λ

Then, (1) ε A0 ≺ λ.  A[B0 ] is ε-dense in Cc (X)⊗  A. (2) The C ∗ -algebra Cb (X) ⊗ max

ε

Proof (1) A defining family ofseminorms for ε is denoted by (εK,ν )(K,ν)∈K× , where for every element z = ni=1 fi ⊗ xi ∈ Cc (X) ⊗ A, we have ' ! n ! ! !

◦

◦ ! ! εK,ν (z) := sup ! φ(fi )y (xi )! : φ ∈ UK , y ∈ Uν .

(8.1.3)

i=1

Note that UK◦ is the polar of the closed unit semiball UK := {f ∈ Cc (X) : qK (f ) ≤ 1} in Cc (X) and Uν◦ the polar of the closed unit semiball Uν := {x ∈ A : pν (x) ≤ 1} in A[τ ]. The normed injective tensor topology λ on Cb (X) ⊗ A[B0 ] is determined by the vector space  norm  · λ (see also long discussion after (8.1.1)), where for every z = ni=1 fi ⊗ xi ∈ Cb (X) ⊗ A[B0 ], ' ! n ! ! !

◦

◦ ! ! zλ := sup ! φ(fi )y (xi )! : φ ∈ UCb (X) , y ∈ UA[B0 ] , i=1 ◦ standing for the polars of the closed unit balls U (C (X)) with UC◦b (X) and UA b b and U (A[B0 ]) of the C ∗ -algebras Cb (X) and A[B0 ], respectively.

8.2 B∗ -Collections in Tensor Product Locally Convex *-Algebras

215

We shall show that for each (K, ν) ∈ K × , εK,ν (z) ≤ zλ , ∀ z ∈ A0 , so the claim (1) will have been proved. According to (8.1.3) it suffices to show that 



U Cb (X) ⊂ UK and U A[B0 ] ⊂ Uν , ∀ (K, ν) ∈ K × .   Since f ∞ = sup qK (f) : K ∈ K , the  first inclusion is obvious. Let now y ∈ U (A[B0 ]). Then, sup pν (y) : ν ∈  ≤ 1, which implies that y ∈ Uν , for all ν ∈ . (2) Since the C ∗ -algebra Cb (X) is commutative, the normed injective tensor topology λ coincides on Cb (X) ⊗ A[B0 ] with the C ∗ -tensor topology max [144, p. 215, Lemma 4.18]. Denote by  · max the C ∗ -crossnorm corresponding  A[B0 ] can to max. From (1), ε A0 ≺ max. Hence, A0 = Cb (X) ⊗ max

 A. Moreover, by the be continuously embedded into C(X, A) = Cc (X)⊗ ε

continuity of the tensor map ⊗, one gets that

ε

ε

 A[B0 ] ⊂ Cc (X)⊗  A, Cc (X) ⊗ A ⊂ Cb (X) ⊗ A[B0 ] ⊂ Cb (X) ⊗ ε

max

max

ε

where the first inclusion follows from the fact that Cb (X), A[B0 ] are sequentially dense in Cc (X), A[τ ] respectively, according to Theorem 4.2.11.   Example 8.1.4 Let A[τ ] be a complete GB∗ -algebra. We shall show that C(X, A) =  A is a GB∗ -algebra. We consider the C ∗ -algebra A0 = Cb (X) ⊗  A[B0 ]. Cc (X)⊗ ε

λ=max

0 [ε], where A0 [ε] A = A Then, from Lemma 8.1.3, ε A0 ≺ λ = max and Cc (X)⊗ ε

has a jointly continuous multiplication (see [111, p. 391, Theorem 1.1], as well the note  at the beginning of Sect. 8.1). So from Corollary 7.1.3, it follows that  A is a GB∗ -algebra over the ε-closed unit ball U (A0 ) of A0 [ · max ]. Cc (X)⊗ ε

8.2 B∗ -Collections in Tensor Product Locally Convex *-Algebras  ] the Let A[τ ] be a locally convex ∗-algebra with identity e. We denote by A[τ completion of A[τ ]. Since, the multiplication in A[τ ] is jointly continuous (as we  ] is a complete locally convex ∗have emphasized at the beginning of Sect. 8.1), A[τ algebra. In what follows, we shall use, for distinction, the symbol  τ for the topology  ]. We denote by B∗ and B∗ the of the complete locally convex ∗-algebra A[τ  A A  ]. respective B∗ -collections of Definition 3.3.1, in A[τ ], respectively A[τ

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8 Applications II: Tensor Products

Proposition 8.2.1 Let A[τ ] be a locally convex ∗-algebra. Then, the following hold: (1) For every B ∈ B∗, the set B ∩ A ∈ B∗A . A  τ τ  τ  ], then B (2) If B ∈ B∗A and B is the closure of B in A[τ ∈ B∗ and B = B ∩A. A

 

Proof It follows easily by the very definitions.

Let now A1 [τ1 ] and A2 [τ2 ] be locally convex ∗-algebras and let τ be a ∗-admissible topology on A1 ⊗ A2 (see Definition 8.1.1). Let B1 ∈ B∗A1 , B2 ∈ B∗A2 and (B1 ⊗ τ

B2 ) the absolutely convex hull of B1 ⊗ B2 in A1 ⊗ A2 . Denote by (B1 ⊗ B2 ) , τ

 τ

respectively (B1 ⊗ B2 ) , the closure of (B1 ⊗ B2 ) in A1 ⊗ A2 , respectively τ

A2 . Then, we have the following A1 ⊗ τ

Theorem 8.2.2 Let A1 [τ1 ], A2 [τ2 ], B1 and B2 be as in the previous paragraph. Then, the collections   τ  τ

(B1 ⊗ B2 ) : Bi ∈ B∗Ai , i = 1, 2 and (B1 ⊗ B2 ) : Bi ∈ B∗Ai , i = 1, 2 , are contained in B∗A1 ⊗A2 and B∗A ⊗ respectively. 1 A2 τ τ

Proof The absolutely convex hull of the set B1 ⊗ B2 is given by 

(B1 ⊗ B2 ) :=



λi (xi ⊗ yi ) : λi ∈ C,

finite



' |λi | ≤ 1 and xi ⊗ yi ∈ B1 ⊗ B2 .

finite

Using the very definitions, the continuity of the tensor map ⊗ and the fact that the ∗-admissible topology τ fulfills the inequality ε ≺ τ ≺ π, we conclude easily that the collection { (B1 ⊗ B2 ) : Bi ∈ B∗Ai , i = 1, 2} consists of absolutely convex, τ

bounded subsets of A1 ⊗ A2 , which contain the identity and are symmetric and τ

idempotent. Due to the continuity of the algebraic operations of the locally convex ∗-algebras under consideration, all of the preceding properties pass to the τ -(resp.  τ -) closure of the sets (B1 ⊗ B2 ), as before. With regard to this, recall also that the τ

closure of a bounded subset of a topological vector space is bounded. Hence, 

τ

(B1 ⊗ B2 ) : Bi ∈ B∗Ai , i = 1, 2 ⊂ B∗A1 ⊗A2 , resp. τ

  τ

(B1 ⊗ B2 ) : Bi ∈ B∗Ai , i = 1, 2 ⊂ B∗A ⊗ . 1  A2 τ

 

8.2 B∗ -Collections in Tensor Product Locally Convex *-Algebras

217

Now from the investigation done in Example 8.1.4, we are led to the formulation of the following Theorem 8.2.3 Let A1 [τ1 ], A2 [τ2 ] be GB∗ -algebras and B10 , B20 maximal members in B∗A1 , B∗A2 , respectively. Let τ be a ∗-admissible topology on A1 ⊗ A2 . The following statements are equivalent: A2 is a GB∗ -algebra. (i) A1 ⊗ τ

(ii) There is a C ∗ -crossnorm  ·  on A1 [B10 ] ⊗ A2 [B20 ], such that A0 :=  A2 [B20 ] is a C ∗ -algebra contained in A1 ⊗ A2 and τ ≺  ·  on A0 . A1 [B10 ] ⊗ ·

τ

Proof (ii) ⇒ (i) From our hypotheses the following facts are true: (1) Ai [Bi0 ], i = 1, 2 are dense in Ai [τi ], i = 1, 2 (see Theorem 4.2.11), (2) the tensor map ⊗ : A1 [τ1 ] × A2 [τ2 ] → A1 ⊗ A2 is continuous and (3) τ ≺  ·  on A0 . Thus, first τ

τ

τ

A2 ,  A2 [B20 ] ⊂ A1 ⊗ A1 ⊗ A2 ⊂ A1 [B10 ] ⊗ A2 [B20 ] ⊂ A1 [B10 ] ⊗ ·

τ

A2 . Secondly (i) follows by applying Theorem 7.1.2. therefore A0 is τ -dense in A1 ⊗ τ

A2 is a GB∗ -algebra. Then, (A1 ⊗ A2 )[B0 ] is a C ∗ (i) ⇒ (ii) Suppose that A1 ⊗ τ

τ

algebra with respect to the gauge function  · B0 , where B0 is the maximal element 0 0 ∗ ∗ in B∗A ⊗  A (see Theorem 3.3.9(2)). If B1 , B2 are maximal elements in BA1 , BA2 1

τ

2

respectively, then it follows from Theorem 8.2.2 that  τ

A2 )[ (B10 ⊗ B20 ) ] ⊂ (A1 ⊗ A2 )[B0 ]. A1 [B10 ] ⊗ A2 [B20 ] ⊂ (A1 ⊗ τ

τ

So we may restrict  · B0 to A1 [B10 ] ⊗ A2 [B20 ] and obtain A2 )[B0 ].  A2 [B20 ] ⊂ (A1 ⊗ A0 := A1 [B10 ] ⊗ ·B0

τ

(8.2.4)

A2 is a GB∗ -algebra over B0 , we have Thus, A0 is a C ∗ -algebra and since A1 ⊗ τ

A2 )[B0 ], hence on A0 as well. Consequently, the proof is τ ≺  · B0 on (A1 ⊗ τ

completed by considering  ·  =  · B0 .

 

The following example demonstrates that not every complete tensor product of GB∗ algebras under some ∗-admissible topology is necessarily a GB∗ -algebra.

Example 8.2.4 Let A1 and A2 be noncommutative C ∗ -algebras. Consider the injective tensor norm ·λ on A1 ⊗A2 . Then, ·λ is not a C ∗ -norm on A1 ⊗A2 (see

218

8 Applications II: Tensor Products

A2 is not a C ∗ -algebra, but [144, Theorems IV.4.14 and IV.4.19]). Therefore, A1 ⊗ λ

A2 is not a GB∗ -algebra. a Banach ∗-algebra. Hence, by Corollary 3.3.11(i), A1 ⊗ λ

Thus, we are led to the following Definition 8.2.5 When the equivalent conditions of Theorem 8.2.3 hold, the locally A2 will be called a tensor product GB∗ -algebra of the GB∗ convex ∗-algebra A1 ⊗ τ

algebras A1 [τ1 ] and A2 [τ2 ]. The results of Sect. 8.2 lead to the following Question It would be interesting and very helpful to know whether there are A2 could conditions under which the structure of a tensor product GB∗ -algebra A1 ⊗ τ

be described by the collection   τ

(B1 ⊗ B2 ) : Bi ∈ B∗Ai , i = 1, 2 . Maybe this would contribute to a relationship, like the one in Example 8.4.2(3),  Ab , which is nicer than the one we have up to now between Mn (A)b and Mn (C) ⊗ for the C ∗ -algebras

max



 A2 [B20 ] and A1 ⊗ A2 [B0 ], A1 [B10 ] ⊗ ·

τ

A2 , as proof of Theorem 8.2.3 shows (see whose τ -closures coincide in A1 ⊗ τ

also (8.3.5)).

8.3 Tensor Product GB*-Algebras In this section we verify Definition 8.2.5 in some cases. The reader should not expect an analogy of GB∗ -tensor products with the C ∗ -tensor products. For instance, a GB∗ -algebra attains unbounded ∗-representations (see Theorem 6.3.5), so to define the C ∗ -crossnorms max and min on the algebraic tensor product of two such algebras, seems not attainable. On the other hand, the biprojective tensor locally convex topology ε plays, in a way, the role of max (and min) in the GB∗ -case, as Theorem 8.3.2 and Corollary 8.3.6, of this section, show. For instance, suppose that A1 [τ1 ], A2 [τ2 ] in Theorem 8.3.2 are pro-C ∗-algebras (hence GB∗ -algebras), with one of them being commutative. Then, the assumptions of Theorem 8.3.2 are A2 is a pro-C ∗-algebra (hence a GB∗ -algebra), with ε = ω = α satisfied and A1 ⊗ ε

(see (8.1.2) and [60, Corollary 31.16]). A crucial property for the last equalities is the commutativity of one of the pro-C ∗ -algebras A1 [τ1 ], A2 [τ2 ]; the respective classical C ∗ -algebra result, in this case, is Lemma 4.18, p. 215, in [144].

8.3 Tensor Product GB*-Algebras

219

Furthermore, the fact that commutativity appears for one of the factors of a GB∗ tensor product in the results of this section, is not surprising, since the definition of a GB∗ -algebra requires symmetry (see Definition 3.3.2). More precisely, when a GB∗ -algebra A[τ ] is also m-convex, then symmetry means that every element of the form e + x ∗ x, x ∈ A, is invertible in A and to pass symmetry to the topological tensor product of two complete symmetric m∗ -convex algebras, you need one of them to be commutative [60, Corollary 34.15] (the corresponding Banach algebra result is due to K.B. Laursen [109, Theorem III.3, p. 65]). Yet, as we shall see, the C ∗ -crossnorms max, min are essentially used in the tensor product of the C ∗ -algebras generated by the maximal elements of the B∗ collections for the GB∗ -algebras participating in a GB∗ -tensor product algebra (concerning these C ∗ -algebras, see also Remark 3.5.7 and the discussion before it, as well as Remark 3.5.5(4)). Recall that given a GB∗ -algebra A[τ ], the C ∗ -algebra A[B0 ] generated by the maximal element B0 of the collection B∗A is the pillar on which the study of the structure of a GB∗ -algebra is based. Lemma 8.3.1 that follows is required in the proof of Theorem 8.3.2. Before we proceed to its statement, we set some notation that we shall need for it. Let A1 [τ1 ], A2 [τ2 ] be two locally convex ∗-algebras and τ a ∗-admissible topology on A1 ⊗ A2 (see Definition 8.1.1). Suppose that 1 = {p}, 2 = {q} and ⊗τ = {τp,q } are the defining families of ∗-seminorms for the topologies τ1 , τ2 , τ respectively. To avoid indices in the preceding families 1 , 2 and ⊗τ , we abuse the symbol D(p ) as in (3.5.24) and put  D( 1 ) := x ∈ A1 : sup p(x) < ∞ , p∈ 1

 D( 2 ) := y ∈ A2 : sup q(y) < ∞ , q∈ 2

 A2 : D( ⊗τ ) := z ∈ A1 ⊗ τ

sup

(p,q)∈ 1 × 2

τp,q (z) < ∞ .

In this regard, we have Lemma 8.3.1 Let A1 [τ1 ] and A2 [τ2 ] be locally convex ∗-algebras and τ a ∗admissible topology on A1 ⊗ A2 . Let 1 , 2 , ⊗τ be as before. Suppose that for each p, q, the ∗-seminorm τp,q is a cross-seminorm, i.e., τp,q (x ⊗ y) = p(x)q(y), for all x ∈ A1 and y ∈ A2 . Then, D( 1 ) ⊗ D( 2 ) ⊆ D( ⊗τ ), γ

where γ is the normed projective tensor topology. . Proof According to our modified notation just before Lemma 8.3.1, we use in the place of p as in (3.5.24), the symbols p i , i = 1, 2 and p ⊗τ . Namely, p 1 (x) :=

220

8 Applications II: Tensor Products

supp∈ 1 p(x), x ∈ D( 1 ), p 2 (y) := supq∈ 2 q(y), y ∈ D( 2 ) and p ⊗τ (z) := sup(p,q)∈ 1× 2 τp,q (z), z ∈ D( ⊗τ ), respectively.  Let now z = ni=1 xi ⊗ yi ∈ D( 1 ) ⊗ D( 2 ). Then, we have p ⊗τ (z) := ≤ ≤

sup

(p,q)∈ 1 × 2

sup

τp,q (z) =

n 

(p,q)∈ 1 × 2 i=1 n 

sup

(p,q)∈ 1× 2

) xi ⊗ yi

i=1

τp,q (xi ⊗ yi )

sup p(xi ) sup q(yi ) =

i=1 p∈ 1

τp,q

( n 

q∈ 2

n 

p 1 (xi )p 2 (yi )| < ∞.

i=1

Taking now infimum with respect to all representations of z in D( 1 ) ⊗ D( 2 ), we obtain p ⊗τ (z) :=

sup

(p,q)∈ 1 × 2

τp,q (z) ≤ zγ , ∀ z ∈ D( 1 ) ⊗ D( 2 ),

where the mentioned infimum is denoted by  · γ (see also long discussion after (8.1.1)) and reads as follows zγ := inf

" n

# p 1 (xi )p 2 (yi ), z ∈ D( 1 ) ⊗ D( 2 ) .

i=1

The normed projective tensor topology γ on D( 1 )⊗D( 2 ) is generated by  · γ . It follows that D( 1 ) ⊗ D( 2 ) is embedded continuously in D( ⊗τ )   γ

Theorem 8.3.2 Let A1 [τ1 ] and A2 [τ2 ] be GB∗ -algebras, with a natural family of ∗-seminorms (take, for instance, A1 [τ1 ] and A2 [τ2 ] to be C ∗ - like locally convex ∗- algebras; see Corollary 3.5.4 and Remark 3.5.7(4)). If either A1 or A2 is A2 is a tensor product GB∗ -algebra. commutative, then A1 ⊗ ε

A2 [B20 ] is a C ∗ -algebra Proof Since one of the A1 , A2 is commutative, A1 [B10 ]⊗ λ

A2 [B20 ] can be algebraically embed[144, Theorem IV.4.14]. Suppose that A1 [B10 ]⊗ λ

A2 . By the same proof as that of Lemma 8.1.3(1), ε A [B 0 ]⊗ ded into A1 ⊗  A2 [B 0 ] ≺ 1 ε

1

A2 is a tensor product GB∗ -algebra. λ, therefore by Theorem 8.2.3, A1 ⊗ ε

λ

2

8.3 Tensor Product GB*-Algebras

221

A2 [B20 ] can be algebraically So all that remains is to prove that A1 [B10 ]⊗ λ

A2 . By Lemma 8.3.1, A1 [B10 ] ⊗ A2 [B20 ] ⊂ D( ⊗ε ), where embedded into A1 ⊗ ε

 A2 : D( ⊗ε ) = z ∈ A1 ⊗ ε

p ⊗ε (z) :=

sup

(p,q)∈ 1 × 2

sup

(p,q)∈ 1 × 2

εp,q (z) < ∞ and

εp,q (z), z ∈ D( ⊗ε ).

We may restrict p ⊗ε to A1 [B10 ] ⊗ A2 [B20 ]. Let z0 = A2 [B20 ]. Then, p ⊗ε (z0 ) =

sup

(p,q)∈ 1 × 2

n

i=1 xi

⊗ yi ∈ A1 [B10 ] ⊗

εp,q (z0 )

! ! n ' ! ! ! ! = sup sup ! x (xi )y (yi )! : x ∈ Up◦ , y ∈ Uq◦ ! ! (p,q)∈ 1 × 2 i=1 ! ' ! n ! ! ! !



◦

◦ = sup ! x (xi )y (yi )! : x ∈ UA [B 0 ] , y ∈ UA [B 0 ] 1 1 2 2 ! ! i=1

= z0 λ , where U ◦

, U ◦ 0 are the polars of the closed unit balls U (A1 [B10 ]), A1 [B10 ] A2 [B2 ] U (A2 [B20 ]) of the C ∗ -algebras A1 [B10 ], A2 [B20 ], respectively. A2 is complete, it follows from the same proof as in [60, Theorem Since A1 ⊗ ε 10.23] that D( ⊗ε ) is complete with respect to p ⊗ε . Hence, A2 [B20 ] = A1 [B10 ] ⊗  A2 [B20 ] ⊂ A1 ⊗ A2 . A1 [B10 ]⊗ p ⊗ ε

λ

ε

  As in the case of Theorem 8.2.3, for the GB∗ -algebras A1 [τ1 ] and A2 [τ2 ] of Theorem 8.3.2, one concludes that ε

ε

 A2 [B20 ] = (A1 ⊗ A2 )[B0 ] = A1 ⊗ A2 . A1 [B10 ] ⊗ λ=max

ε

ε

(8.3.5)

Corollary 8.3.3 Let A1 [τ1 ] and A2 [τ2 ] be C ∗ -like locally convex ∗-algebras, with either A1 , or A2 being commutative. Then, the following statements are equivalent: A2 admits a C ∗ -like family of seminorms defining the topology ε. (i) A1 ⊗ ε

A2 is a C ∗ -like locally convex ∗-algebra. (ii) A1 ⊗ ε

222

8 Applications II: Tensor Products

Proof It is obvious that (ii) ⇒ (i). So we prove that (i) ⇒ (ii). Since A1 [τ1 ] and A2 [τ2 ] are C ∗ -like locally convex ∗-algebras, they are also GB∗ -algebras and A1 [B10 ] = (A1 )b , A2 [B20 ] = (A2 )b , according to Theorem 3.5.3 and A2 is a tensor product GB∗ -algebra. Remark 3.5.7(3). Thus, by Theorem 8.3.2, A1 ⊗ ε

Hence, from the proof (i) ⇒ (ii) of Theorem 8.2.3, the relation (8.2.4) holds, with B0 0 0 the maximal element in B∗A ⊗  A ‘ Since either A1 [B1 ] or A2 [B2 ] is commutative, 1

ε

2

we conclude that  · B0 =  · λ = p ⊗ε , on A0 := A1 [B10 ]⊗A2 [B20 ], (see [144, p. 215, Lemma 4.18] and the proof of Theorem 8.3.2). It follows that B0 coincides with the unit ball U (p ⊗ε ) of the C ∗ -algebra D( ⊗ε ) (for the last two symbols, see discussion after (3.5.24), before Lemma 3.5.2 and the beginning of the proof of Lemma 8.3.1). So from (i) and Theorem 3.5.3, we obtain (ii).    A of Example 8.3.4 Consider the locally convex ∗-algebra C(X, A) ∼ = Cc (X)⊗ ε

Example 8.1.4, where X is a locally compact space and A a C ∗ -like locally convex ∗-algebra. Recall that if {pν }ν∈ is a C ∗ -like family of seminorms for A, a defining family of ∗-seminorms for C(X, A) is given by the functions   qK,ν (f ) := sup pν (f (x)), x ∈ K , ∀ f ∈ C(X, A), for all ν ∈  and K ⊂ X, compact. It is easily verified that, for every compact subset K of X and every ν ∈ , there exists ν ∈  such that qK,ν (fg) ≤ qK,ν (f )qK,ν (g), qK,ν (f )2 ≤ qK,ν (f ∗ f ) and qK,ν (f ∗ ) ≤ qK,ν (f ),

for all f, g ∈ C(X, A), i.e., the family of seminorms (qK,ν )ν∈ is a C ∗ -like family of seminorms defining the topology of C(X, A). Hence, by Corollary 8.3.3, C(X, A) is a tensor product C ∗ -like locally convex ∗-algebra. In [59, p. 241, Corollary] it is proved that if A[τ ] is a locally convex algebra with jointly continuous multiplication and an identity element e, then there is a family of seminorms, say {q}, defining the topology τ of A, such that q(e) = 1, for every q. Suppose now that A[τ ] has a continuous involution ∗. Then, a family of ∗seminorms {q } is defined on A that preserves involution, identity and defines the topology τ . Indeed, using the preceding family of seminorms {q}, we put q (a) := max{q(a), q(a ∗)}, a ∈ A. It follows that for every q , q (a ∗ ) = q (a), for all a ∈ A and q (e) = 1. So the aforementioned result in [59], for the ∗-case, reads as follows: Lemma 8.3.5 Let A[τ ] be a locally convex ∗-algebra with identity e and jointly continuous multiplication. Then, τ is defined by a family = {q } of ∗-seminorms, such that q (e) = 1, for every q ∈ .

8.4 GB*-Nuclearity

223

Corollary 8.3.6 Let A1 [τ1 ] and A2 [τ2 ] be complete locally convex ∗-algebras with identities e1 , e2 , respectively. Consider the following statements: A2 is a tensor product GB∗ -algebra. (i) A1 ⊗ ε

(ii) A1 [τ1 ] and A2 [τ2 ] are GB∗ -algebras. Then (i) implies (ii). If A1 [τ1 ] and A2 [τ2 ] are GB∗ -algebras with a natural family of ∗-seminorms and either A1 or A2 is commutative, then (ii) implies (i) too. Proof (i) ⇒ (ii) Let 1 = {p} (resp. 2 = {q}) be a family of ∗-seminorms defining the topology τ1 on A1 (resp. τ2 on A2 ). Denote by p,q = {εp,q } the family of ∗-seminorms defining the topology ε on A1 ⊗A2 . Then, εp,q (x ⊗y) = p(x)q(y) for all x ∈ A1 and y ∈ A2 . Applying Lemma 8.3.5, we may suppose that the families of ∗-seminorms 1 , respectively 2 , preserve the corresponding identities. Thus, one has εp,q (x ⊗ e2 ) = p(x)q(e2 ) = p(x), ∀ x ∈ A1 . Consider now the embedding A2 : x → x ⊗ e2 . A1 !→ A1 ⊗ ε

A2 with e1 ⊗ e2 ∈ A1 . Hence, Then A1 [τ1 ] becomes a closed ∗-subalgebra of A1 ⊗ ε

by Proposition 3.3.19, A1 [τ1 ] is a GB∗ -algebra. Similarly, A2 [τ2 ] is a GB∗ -algebra. (ii) ⇒ (i) It follows from Theorem 8.3.2.   Remark 8.3.7 Note that for Ai [τi ], i = 1, 2, as before and τ a ∗-admissible A2 is a tensor product GB∗ -algebra, one topology on A1 ⊗ A2 , such that A1 ⊗ τ

concludes from the proof of Corollary 8.3.6, that A1 [τ1 ] and A2 [τ2 ] are GB∗ algebras too.

8.4 GB*-Nuclearity The concept of a nuclear topological vector space was first given by A. Grothendieck and much of the theory of these types of spaces was developed by him and published in [69]. The nuclearity of a C ∗ -algebra was first discussed under the term “Property T” by M. Takesaki, in 1964 (see, for instance, [144, p. 204, Vol. III]). The present term is due to E. Effros and C. Lance [55]. Nuclear C ∗ -algebras are developed in various directions and are very important in the C ∗ -algebra theory; see also [39, 41, 101, 102] and the literature therein. For the non-normed case, see [26, 124].

224

8 Applications II: Tensor Products

Throughout this section, given two GB∗ -algebras A[τi ], i = 1, 2 and their tensor A2 , we shall denote by B0i and B0 the maximal elements product GB∗ -algebra A1 ⊗ τ

respectively. in B∗Ai , i = 1, 2 and B∗A ⊗ 1 A2 τ Our definition of GB∗ -nuclearity that follows is based on the definition of nuclearity given by S.J. Bhatt and D.J. Karia for pro-C ∗ -algebras in [26, Theorem 4.5]. Definition 8.4.1 Let A[τ ] be a GB∗ -algebra with B0 the greatest member in B∗A . We say that A[τ ] is nuclear if the C ∗ -subalgebra A[B0 ] of A[τ ] is nuclear. Examples 8.4.2 (1) Every nuclear pro-C ∗ -algebra A[τ ] is a nuclear GB∗ - algebra. Indeed, by Bhatt and Karia [26, Theorem 4.5], a pro-C ∗ -algebra A[τ ] is nuclear, if and only if, its bounded part Ab (which is a C ∗ -algebra, see Remark 3.5.7(3)) is nuclear. Moreover, A[τ ] is a GB∗ -algebra with B0 the closed unit ball of Ab , therefore Ab = A[B0 ] (see Example 3.3.16(1)), whence the conclusion. (2) Every commutative GB∗ -algebra is nuclear. In particular, the Arens algebra Lω [0, 1] (see Example 3.3.16(5)) is a nuclear GB∗ -algebra, with (Lω [0, 1])[B0 ] = L∞ [0, 1]. (3) Let A[τ ] be a commutative C ∗ -like locally convex ∗-algebra and Mn (A) all n × n matrices with entries from A. The assumption for A[τ ] implies that this is a GB∗ -algebra with A[B0 ] = Ab , B0 being the maximal element in B∗A . Then, since Mn (C) is nuclear, all C ∗ -crossnorms coincide on Mn (C) ⊗ Ab and are also equal to the injective norm λ since Ab is commutative (see [144, A Theorem IV.4.14]). Thus, Theorem 8.3.2 implies that Mn (A) ∼ = Mn (C)⊗ ε

is a GB∗ -algebra. Let us now consider the structure of Mn (A) as a locally convex ∗-algebra through that of A[τ ]. If τ is defined by the family = {p} of seminorms, then a defining family of seminorms, say {rp, ˜ q˜ }, p, q ∈ , for Mn (A), is given as follows: Let M := An be the finitely generated free (left) A-module, corresponding to A. Then, each x ∈ Mn (A) defines a continuous A-linear operator Tx on M given by Tx (a) :=

n  i=1

x1i xi , . . . ,

n 

 xni xi ,

i=1

 where xij are the entries of the matrix x and a = ni=1 xi ei is an element of M, with xi ∈ A and ei := (δij )1≤j ≤n ∈ An , such that δii = e (the identity of A) and δij = 0, for i = j . Furthermore, since is a C ∗ -like family, for every p ∈ , there is q ∈ , such that p(xy) ≤ q(x)q(y), for any x, y ∈ A. For such

8.4 GB*-Nuclearity

225

p, q as before, we put   rp, ˜ x (a)) : q(a) ˜ ≤ 1 , x ∈ Mn (A), where ˜ q˜ (x) := sup p(T p(T ˜ x (a)) :=

n n n     p xj i xi and q(a) q(xi ), ∀ a ∈ M ˜ := j =1

i=1

i=1

(see also [60, pp. 109–110, (6)]). An easy calculation shows that rp, ˜ q˜ (x) < ∞, for all x ∈ Mn (A). Also, we can easily verify that Mn (A)b = Mn (Ab ) =  Ab , where the last equality results from the nuclearity of Mn (C). At Mn (C) ⊗ max

the same time, we conclude that Mn (A)b is nuclear, therefore what remains to be proved is that Mn (A)[B0 ] = Mn (A)b . This will follow by showing that Mn (A) accepts a C ∗ -like family of seminorms defining its topology, so that by Corollary 8.3.3, Mn (A) will be a C ∗ -like locally convex ∗-algebra. This demands a series of technical long computations, which are based on the same method, with obvious modifications, as the one applied in [110, Section 2, pp. 463 - 467] for showing that: if B[τ ] is a pro-C ∗-algebra and Mn (B) the algebra of all n × n matrices with entries in B, then Mn (B) also becomes a pro-C ∗ algebra (ibid., p. 466, Corollary 2.1). For this, we use the form of the seminorms ∗ {rp, ˜ q˜ } defined above and the fact that = {p} is a C -like family of seminorms ∗ defining the topology τ of our commutative C -like locally convex ∗-algebra A. Note that we can also define a GB∗ -algebra to be nuclear by assuming no identity; this is, for instance, required for the two-sided ideal I in the following Theorem 8.4.3 Let A[τ ] be a GB∗ -algebra. Let I be a closed two-sided ideal in A[τ ]. Then, A[τ ] is nuclear, if and only if, I and A/I are nuclear. Proof By Theorem 7.4.8 we have that I is a ∗-ideal, therefore a GB∗ -subalgebra of A[τ ] by Proposition 3.3.19. Applying again Theorem 7.4.8, we conclude that A/I is a GB∗ -algebra with the quotient topology and (A/I )[BI0 ] = A[B0 ]/(I ∩ A[B0 ]), where BI0 , B0 are the maximal elements in B∗A/I , B∗A , respectively. Now, A[τ ] is nuclear, if and only if, A[B0 ] is nuclear, if and only if, (see [39, Corollary 4]) I ∩A[B0 ] and A[B0 ]/(I ∩A[B0 ]) are nuclear, if and only if, (from the previous paragraph), I ∩ A[B0 ] and (A/I )[BI0 ] are nuclear. But, the GB∗ -structure of I is determined by the collection B∗I = I ∩ B∗A , so that I [I ∩ B0 ] = I ∩ A[B0 ]. Thus, we have proved that the GB∗ -algebra A[τ ] is nuclear, if and only if, I and A/I are nuclear GB∗ -algebras.   Applying Theorem 8.4.3 and the open mapping theorem for Fréchet spaces, we obtain the following Corollary 8.4.4 Let A1 [τ1 ], A2 [τ2 ] be Fréchet GB∗ -algebras. If φ : A1 [τ1 ] → A2 [τ2 ] is a continuous surjective ∗-homomorphism, then A1 [τ1 ] is nuclear, if and only if, ker(φ) and A2 [τ2 ] are nuclear.

226

8 Applications II: Tensor Products

Theorem 8.4.5 Let A[τ ] be a GB∗ -algebra. Then, the following are equivalent: (1) A[τ ] is nuclear.  B = A[B0 ] ⊗  B, with max = min. (2) For every C ∗ -algebra B, A[B0 ] ⊗ max

min

B = A[B0 ]⊗ B, with α = ω, where (3) For every pro-C ∗ -algebra B[τB ], A[B0 ]⊗ α

ω

α, ω are, the corresponding to the classical C ∗ -cross-norms  · min ,  · max , in the context of pro-C ∗ -algebras (see (8.1.2)). Proof The equivalence of (1), (2) follows easily from the very definitions. (2) ⇒ (3) The topology τB is determined by an upwards directed family of C ∗ −seminorms, say, {q}. Then, B[τB ] = lim Bq , where each Bq = B/ ker(q), ← −  Bq = is a C ∗ −algebra [60, pp. 15–16]. Thus, from (2), we have that A[B0 ] ⊗ max

 Bq , with max = min, for each C ∗ −seminorm q. The assertion now follows A[B0 ] ⊗ min

from [60, Corollary 31.11]. (3) ⇒ (2) It is obvious.

 

Proposition 8.4.6 Let Ai [τi ], i = 1, 2, be nuclear GB∗ -algebras. Suppose A2 is the tensor product GB∗ -algebra of Ai [τi ], i = 1, 2, such that that A1 ⊗ τ

A2 )[B0 ] = A1 [B01 ] ⊗ A2 is nuclear too.  A2 [B02 ]. Then, A1 ⊗ (A1 ⊗ τ

·B0

τ

Proof It is straightforward from the nuclearity of the C ∗ −algebras Ai [B0i ], i = 1, 2 and the fact that max =  · B0 = min (see [102, p. 389]).   Under the basis put in this section, it would be interesting to see how far GB∗ -nuclearity goes. A natural question is, for instance, under what conditions a subalgebra of a nuclear GB∗ -algebra is also nuclear. A further investigation of GB∗ -nuclearity was started by the third author in [152], where a more detailed study of completely positive maps of GB∗ -algebras and their relationship to GB∗ -nuclearity is carried out.

8.5 Some Applications on Tensor Product GB*-Algebras For the terminology used in what follows, cf. the discussion in Chap. 5, before Definition 5.1.1 and after Proposition 5.1.2. Let A be a ∗-algebra and p an unbounded C ∗ -(semi)norm of A with domain D(p) (see discussion before Definition 3.5.1). Denote by Ap the C ∗ -algebra completion of the pre-C ∗ -algebra (D(p)/Np )[ · p ] under the C ∗ -norm x + Np p := p(x), x ∈ D(p), induced by p, where Np = {x ∈ D(p) : p(x) = 0}. For the concepts of a w-semifinite unbounded C ∗ -seminorm and of a well-behaved ∗-representation of a ∗-algebra, that we use right after, we refer the reader to [24]. Let now A[τ ] be a GB∗ -algebra, π a ∗-representation of A in a Hilbert space H, with domain D(π), and IbA = {x ∈ B(A) : ax ∈ B(A), ∀ a ∈ A}. For the

8.5 Some Applications on Tensor Product GB*-Algebras

227

∗-subalgebra B(A), see Remark 3.5.5(4). For every x ∈ IbA , define a C ∗ -seminorm on π(A) as follows px (π(a)) = π(x ∗ ax), ∀ a ∈ A. The locally convex topology induced by the family of C ∗ -seminorms {px : x ∈ IbA }, we denote with τlu [22, p. 96]. A ∗-representation π of a ∗-algebra A is said to be uniformly nondegenerate if [π(IbA )D(π)] = Hπ , where [K] means the closed linear span of a subset K of a Hilbert space H. In this regard, we now have the following Proposition 8.5.1 Let A1 [τ1 ] and A2 [τ2 ] be metrizable C ∗ -like locally convex ∗algebras with an identity element. Suppose that either A1 or A2 is commutative. Then, every uniformly nondegenerate ∗-representation π of the tensor product A2 into the image I m(π)[τlu ] of π, under the topology τlu , is GB∗ -algebra A1 ⊗ ε

continuous. The restrictions π1 , π2 of π to A1 [τ1 ], A2 [τ2 ], respectively, are also continuous. Proof From Theorem 3.5.3, every C ∗ -like locally convex ∗-algebra is a GB∗ A2 is a GB∗ -algebra. The latter algebra, therefore Theorem 8.3.2 yields that A1 ⊗ ε

being Fréchet is quasi-complete and bornological [74, p. 222, Proposition 3] (for the preceding terminology see also Definition 4.3.14(2) and the discussion after it). So the result follows from [22, Proposition 4.8].   Notice that if π is an unbounded ∗-representation of a Fréchet ∗-algebra A[τ ], then there exist two topologies on I m(π) making π continuous [136, Theorem 3.6.8]. These topologies τi , i = 1, 2, do not necessarily have the property that I m(π)[τi ] is complete [136, Proposition 3.3.19(i)]. Proposition 8.5.2 Let A1 [τ1 ] and A2 [τ2 ] be metrizable C ∗ -like locally convex ∗algebras with an identity element. Suppose that either A1 or A2 is commutative. A2 is continuous. Then, every positive linear functional on A1 ⊗ ε

A2 is a quasi-complete and Proof As in the proof of Proposition 8.5.1, A1 ⊗ ε

bornological GB∗ -algebra, so the result follows from Corollary 7.2.2.

 

A consequence of Theorem 8.3.2 and of the proofs of Proposition 8.5.1 and Proposition 8.5.2 is the following Corollary 8.5.3 Let A be a Fréchet GB∗ -algebra with A[B0 ] = D(p ) (see (3) and (4) in Remark 3.5.7). Then, we have that (1) every uniformly nondegenerate ∗-representation  A → I m(π)[τlu ] π : C(R, A) ∼ = Cc (R)⊗ ε

is continuous;

228

8 Applications II: Tensor Products

A is continuous. (2) every positive linear functional on C(R, A) ∼ = Cc (R)⊗ ε

C ∗ -like

locally convex ∗-algebra with Proposition 8.5.4 Let A[τ ] be a metrizable B0 = U (Ab ) (the closed unit ball of Ab ) and IbA = {0}. Consider the Fréchet GB∗ algebra C(R, A) of all A-valued continuous functions on the real line R. Then, A attains a well-behaved ∗-representation. C(R, A) ∼ = Cc (R)⊗ ε

Proof By Theorem 3.5.3 and Proposition 3.5.8, we have Ab [ · b ] = A[B0 ][ · B0 ] = B(A)[ · B0 ].

(8.5.6)

Hence, Ab [ · b ] is a C ∗ -algebra, therefore hermitian (see discussion after Definition 3.1.2). In the case of the Fréchet pro-C ∗-algebra (hence Fréchet GB∗ -algebra) Cc (R) we have that Cc (R)[·b ] = Cb (R), the C ∗ -algebra of all bounded continuous functions on R, and IbCc (R) = {f ∈ C(R) : supp(f ) is compact}. Consequently, C (R) = {0} = ker( · ∞ ) (see also [22, Example 3.9(3)]). We can now apply Ib c [61, Theorem 5.2(2) and Application (1) in p. 269] to obtain a well-behaved ∗representation of the tensor product GB∗ -algebra C(R, A) (Example 8.1.4).   An example of a GB∗ -algebra A[τ ] with the property IbA = {0} can be found in [25, Example 6.4]. In general, for i = 1, 2 and any metrizable C ∗ -like locally A convex ∗-algebras Ai [τi ], such that at least one of them is commutative and Ib i = ∗ A2 (cf. Theorem 8.3.2) {0}, we obtain that the tensor product GB -algebra A1 ⊗ accepts a well-behaved ∗-representation.

ε

Notes Lemma 8.3.5 is a *-version of a result of A. Fernández and V. Müller, which can be found in [59, p. 241, Corollary]. Definition 8.4.1 is an analogue of a definition of nuclearity for pro-C ∗ -algebras given by S.J. Bhatt and D.J. Karia in [26, Theorem 4.5]. All other results are due to the first three authors of this book and are contained in [64].

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Index

(E, ), 178 (B)A , 16 (A, ), 178 (A+ ) , 150 (A0 )h , 75 (A0 )b , 118 B(A), 75 B0 (τ ), 91 C , 181 D(A, x), 180 D(A[C ], I d), 182 E

, 179 EW∗ -algebra left, 122 right, 122 Fx , 30 Fx , 30 GB∗ -algebra commutative functional calculus in a, 80 equivalent topologies on a, 91 functional calculus in a, 128 topological, 52 GB∗ -algebra of Dixon, 53 GB∗ -algebra over B0 , 52 GB∗ -algebra, without identity, 54 GNS-construction, 138 q GA , 10 GA , 9 H (A), 40 I d, 182 J , 122 JA , 48 L(E), 179 L∞ (τ0 ), 123

L∞ (ϕ0 ), 118 Lp (τ0 ), 123 Lp (ϕ0 ), 118 Lp -norm of an operator in Lp (ϕ0 ), 118 p L2 (A0 ), 118 L∞ 2 (τ0 ), 124 Lω2 (A0 ), 119 Lω2 (τ0 ), 124 N(A), 40 Rk (Ai ), k = p, q, i = 1, 2, 213 Rλ , 21 Rλ (x), 21 A2 ), 213 Rp,q (A1 ⊗ π

S(A), 180 Ta , 180 1 ), 72 U (p U (p ), 71  U π(A) , 139 V (T ), 179 V (A, a), 180 V (A[C ], T ), 182 V (A[C ], T ), 182 W (T ), 179 W (a), 179 W 1 (T ), 179 Wq1 (T ), 179 W 2 (T ), 179 W (T ), 179 W (a), 179 Wp1 (T ), 179 α, 212 αp,q (·), 213 ≈, 9 β(·), 17 ◦, 10

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Fragoulopoulou et al., Generalized B*-Algebras and Applications, Lecture Notes in Mathematics 2298, https://doi.org/10.1007/978-3-030-96433-7

235

236 ∼ =, 41 γ , 212, 219 λ, 212, 214 B, 178 C∗ , 20 K, 98 N, 98 M[M0 ], 123 M[U0 (M0 )], 118 B, 16 B0 ≡ (B0 )A , 11 B1 , 12 H, 178 H , 180 M0 , 32 μ(T ), 123 μ0 (T ), 118 , 201 ω, 212 ωp,q (·), 213 ⊕i∈I πi , 138 S, 9 τ S ,9 ∗ W (a) , 186 π, 212 π1 ⊂ π2 , 105 πp,q (·), 212 ≺, 9 || · ||1 , 11 || · ||max , 215 || · ||γ , 212, 220 || · ||∞ , 58, 118 || · ||b , 76 || · ||p , 118 || · ||B , 12 || · ||λ , 212, 214 || · || , 182 || · ||p∧2 , 119 , 69 A T, 104  ∗ , 196 B A [τ ]π , 165 A b (π), 105 D (M), 104 D , 104 M ρ(·), 20 ρA (·), 20 σ (·), 20 σ (A, A ), 21 σ D (·), 66 σB (·), 197 σC (·), 184 D (·), 66 σA σ (·), 185

Index σA (·), 20 ∼, 91 τ ∼ τ , 91 τ (E, E ), 95 τ $ , 96 τ u , 108 τs∗ , 107 τlc∗ , 96 τ pf , 92 τ∗u , 108 τ1 , 11 τ2ω , 124 τω , 125 τs , 107 τu , 108 τK , 96 τw , 107 τp,q (·), 219 τqu , 109 ∗ , 109 τqu

-contraction, 181

, 178

, A , 10

⊗α , 213

⊗ω , 213

⊗π , 212

⊗τ , 219

⊗ε , 212 , 179 p , 178 q , 179 εK,ν (·), 214 ε, 212 εp,q (·), 212  τ , 215 † , 103 e1 ≡ (0, 1), 10 1 (·), 72 p pM (·), 108 pξ (·), 107 p∗M (·), 108 ξ p∗ (·), 107 pM (·), 108 pTM (·), 109 ξ pT (·), 107 p (·), 57, 71 pξ (·), 125 ξ p∗,T (·), 107 M (·), 109 p∗,T p ⊗τ , 220 pA (·), 49 qK (·), 95, 214 qB (·), 178

Index

237

qK,ν (·), 214 r(·), 27 rπ (·), 165 rA (·), 27 sp(·), 20 spA (·), 20 tπ , 105 tM , 104 x > 0, 42 x ≥ 0, 42 x ◦ , 10 ∗-algebra, 40 Banach, 40 Fréchet, 40 locally convex, 40 C ∗ -like, 71 hermitian, 43 symmetric, 44 normed, 40 topological, 40 hermitian, 66 symmetric, 52 ∗-algebra of functions, 68 ∗-antirepresentation, 117 ∗-representation closed integrable, 105 self-adjoint, 105 ∗-subalgebra, 40 C, complex numbers, 9 N, natural numbers, 9 R, real numbers, 9 A[B], 11 D (p ), 71 A , 92 A[B0 ], 14, 52 A[C ], 181 A[|| · ||], 10 A[τ ], 10 A+ , 42, 82, 130 A∗ , 92 Apf , 92  A2 , 211 A1 ⊗ τ

A1 ⊗ A2 , 212 τ

A1 , 10 A1 [|| · ||1 ], 11 A1 [τ1 ], 11 Aρ , 33 Ab , 76 A0 , 11  A, 214 C (X, A) = Cc (X)⊗ C0 (S), 79 C1 (S), 81

ε

D (π), 105 D (p ), 57 D (p )1 , 71 L† (D ), 103 Mb , 114 [τ ]), 163 P (A P (A), 92 U0 (A0 ), 116 V0 (A0 ), 118

A0 , 117 B∗A[τ ] , 51 BA , 52 B∗A , 51 (∗-)embedding topological, 41 (∗-)isomorphism topological, 41 [A], 134

Algebra AO ∗ -, 142 C ∗ -convex, 42 C ∗ -, 42 EC∗ -, 114 EW∗ -, 114 standard, 123 GB∗ -, 52, 196 nuclear, 224 semi-, 196 tensor product, 218 HB∗ -, 196 semi-, 196 Q-, 56 σ -C ∗ -, 42 m-convex, 10 m∗ -convex, 42 O ∗ -, 104 O∗ C ∗ -like, 115 closed, 104 closure of an, 104 self-adjoint, 104 Banach, 10 closed O ∗ bounded part of a, 114 integrable, 104 symmetric, 114 hermitian, 42 locally convex, 10 functional calculus in a, 31 hypocontinuous, 180 pseudo-complete, 12 normed, 10

238 pre C ∗ -, 42 pro-C ∗ -, 42 semisimple, 48 spectral, 42 symmetric, 42 topological, 10

Barrel, 97 Basic subcollection, 12 Bornivorous, 98 Bounded part, 76

Carrier space, 32 Cauchy domain, 30 Character, 62 Circle operation, 10 Cone, 146 dual, 146 normal, 146

Element (Allan-)bounded, 11 bounded, 11 dissipative, 187 hermitian, 180 normal, 40 positive, 42, 82, 130, 207 quasi-invertible, 10 self-adjoint, 40 strictly positive, 42 Family of ∗-seminorms natural, 77 Family of seminorms C ∗ -like, 71 M ∗ -like, 70 defining, 11 saturated, 10 First order decay in the right half–plane, 187

Gauge function, 12 Gelfand transform, 65 Gelfand–Naimark type theorem algebraic commutative, 69 algebraic noncommutative, 139 noncommutative, 144

Index Hermitian decomposition, 201 Hilbert algebra, 117 achieved, 118 maximal, 118 pure unbounded, 117 unbounded, 117 Hilbert space direct sum, 138 Hypocontinuous bilinear map, 133

Integral on Lp (ϕ0 ), 118 Involution, 40 hermitian, 42, 43

Jacobson radical, 48

Linear functional positive, 83 ∗-preserving, 84 extendable, 84 strongly, 163 self-adjoint, 83

Max, 212 Min, 212 Minkowski functional, 12

Norm, 10 C ∗ -, 42 m∗ -, 40 algebra, 10 Numerical range of an element, 179 Numerical range of an operator, 179

Operator closable, 104 closed, 104 closure of an, 104 core of an, 104 essentially self-adjoint, 104 extension of an, 104 self-adjoint, 104 symmetric, 104

Index Projection, 120 Property C ∗ -, 42 Property [A], 134 Pták function, 49

Quasi-inverse, 10

Radius of boundedness, 17 Representation ∗-, 105 algebraic direct sum, 138 closed, 105 closure of a, 105 direct sum, 138 domain of a, 105 extension of a, 105 faithful, 105 functional, 65, 68 Gelfand, 65 left regular, 117 right regular, 117 Resolvent function, 21 Resolvent set, 20

Seminorm C ∗ -, 42 unbounded, 71 m-, 10 m∗ -, 42 unbounded, 71 Space k-, 58 barrelled, 97 bornological, 98 Gelfand, 32 Spectral radius, 27

239 Spectrum of an element, 20, 66, 184 Strong product, 105 Strong scalar multiplication, 105 Strong sum, 105 Tensor map ⊗, 211 Topology Lω2 -, 124 associated barrel, 98 associated bornological, 98 graph, 104 Mackey, 95 positive-functional, 92 quasi-strong, 107 quasi-strong∗ , 107 quasi-uniform, 109 quasi-uniform∗ , 109 strong, 107 strong∗ , 107 tensor ∗-admissible, 211 compatible, 212 injective ⇔ biprojective locally convex, 212 injective C ∗ -convex, 212 normed injective, 214 normed projective, 219 projective C ∗ -convex, 212 projective locally convex, 212 uniform, 108 weak, 107 Topology T, 132

Unitization of a topological algebra, 11 Unitization of an algebra, 10

Wedge, 162

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