TOPOLOGICAL ALGEBRAS SELECTED TOPICS

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1 TOPOLOGICAL ALGEBRAS SELECTED TOPICS Anastasios MALLIOS Mathematical Institute University ofathens Greece 1986 NORTH-HOLLAND-AMSTERDAM NEW YORK»OXFORD»TOKYO

2 xiii Contents Preface ix PART I. GENERAL THEORY CHAPTER I. General Concepts 1. Preliminaries. Definitions 1 2. Examples of topological algebras 9 2. (1). The algebra (E) 10 2.(2). The algebra 0S(R) 10 2.(3). The algebra C [t] 11 2.(4). The Arens algebra L ([0, 1]) Topologies defined by submultiplicative semi-norms Continuity of the multiplication. Complete topological algebras Topological algebras admitting locally m-convex topologies (1). Michael's Theorem (2). 4-convex algebras Certain particular classes of topological algebras 39 6.(1). Locally bounded algebras (2). g-algebras (3). Advertibly complete algebras 44 CHAPTER II. Spectrum (Local Theory) 1. Spectrum of an element. Spectral radius The resolvent set Topological algebras with continuous inversion Waelbroeck algebras Topological division algebras. Gel'fand-Mazur Theorem Maximal ideals Characters. Closed maximal ideals Appendix: Schur's Lemma 77

3 CONTENTS CHAPTER III. Projective Limit Algebras 1. Initial topologies. Topological subalgebras. Cartesian products Projective Systems of topological algebras Representations of locally m-convex algebras as projective limits. Arens-Michael decomposition Applications of the Arens-Michael decomposition Advertibly complete locally m-convex algebras Spectral properties of advertibly complete locally m-convex algebras 99 CHAPTER IV. Inductive Limit Algebras 1. Inductive Systems of algebras. Algebraic pireliminaries Final topologies. Inductive Systems of topological algebras Inductive limits of locally m-convex algebras Examples of topological inductive limit algebras (1). The algebra K(X) (2). The algebra S)(X) as a topological subalgebra of Cf (X) (3). The algebra 0(K) 134 CHAPTER V. Spectrum (Global Theory) 1. Spectrum of a topological algebra Spectrum of the completion of a topological algebra Spectrum of an inductive limit topological algebra Envelopes of holomorphy The dual of the Arens-Michael decomposition (1). Compactly generated topological Spaces The dual of the Arens-Michael decomposition (contn'd.) Spectrum of a projective limit topological algebra. Dense projective limit algebras Appendix: Generalized spectrum (1). General theory (2). Generalized spectrum of a topological projective limit algebra (3). Generalized spectrum of a topological inductive limit algebra 180

4 CONTENTS XV CHAPTER VI. The Gel'fand Map 1. Continuity of the Gel'fand map Boundaries Functional calculus. Holomorphic functions of a Single element in a topological algebra Functional calculus (contn'd.). Holomorphic functions of finite many elements in a topological algebra Appendix: Generalized Gel'fand map 212 CHAPTER VII. Spectra of Certain Particular Topological Algebras 1. Spectrum of the algebra C (X) Spectrum of the algebra C (%) Spectrum of the algebra 0(X). Stein algebras Spectrum of the algebra L (G) 231 e oo (X)(contn d.). The Nachbin Theorem (necessity) The Nachbin Theorem (sufficiency) Appendix: Variants of Nachbin's Theorem (1). Differentiability of class C- m (2). Complexification 251 CHAPTER VIII. Some Special Classes of Topological Algebras 1. Spectrally barrelled algebras (contn'd.) Nachbin-Shirota algebras Functional representations Topological algebras with a given dual Uniform topological algebras convex algebras (contn'd.) Finitely generated topological algebras Functional calculus (contn'd.). The Silov-Arens-Calderön- Waelbroeck theory Miscellanea (1). F2-algebras (2). Nuclear algebras (3). (Ik) -algebras (4). m-infrabarrelled algebras (5). Gel'fand-Mazur algebras (6). Infra-Ptäk algebras Infinite dimensional holomorphy. Spectra of particular topological algebras 311

5 xvi CONTENTS 10.(1). The algebra of continuous polynomials *r(e) (2). Topological algebras of holomorphic functions on infinite dimensional Spaces (3). Holomorphic functions on infinite dimensional spaces (contn'd.). The algebra H(U)[x ] Convolution algebras of C - functions 325 CHAPTER IX. Structure Theory 1. Maximal ideal space (structure space). /z/c-topology Regulär, normal and Silov algebras Hulls of ideals Hulls of ideals: Regulär, normal and Silov algebras (contn'd.) Further results. The Local Theorem Sets of spectral synthesis. Wiener-Tauber algebras Uniform algebras (contn'd.). Riemann algebras 352 PART II. TOPOLOGICAL TENSOR PRODUCTS CHAPTER X. Topological Tensor Products of Topological Algebras 1. Algebraic preliminaries Topological tensor products of locally convex spaces (1). The projective tensorial topology w (2). The inductive tensorial topology i (3). The biprojective tensorial topology e (4). The e-product of L. Schwartz Topological tensor products of topological algebras. Compatible topologies Tensor products of locally bounded algebras Infinite topological tensor product algebras 383 CHAPTER XI. Topological Tensor Product Algebras. Examples 1. The algebra C a (X,JE) The algebra CJ^fX, E) The algebra CJ (X, E)(contn'd.). Nachbin's Theorem (vectorization) The algebra 0(X, E) The algebra L (G, E) (generalized group algebra) 402 CHAPTER XII. Spectra of Topological Tensor Product Algebras 1. Spectrum of a tensor product of topological algebras

6 COHTENTS XVii (numerical case) Spectrum of an infinite topological tensor product algebra Generalized spectrum of a tensor product of topological algebras. Canonical decomposition Inductive limits and generalized spectra Generalized spectra and "point evaluations" 429 CHAPTER XIII. Properties of Permanence of Topological Tensor Product Algebras 1. Boundaries of topological tensor product algebras Continuity of the Gel'fand map Spectrally barrelled algebras Semi-simplicity Identity elements Regularity. Silov algebras Wiener-Tauber condition Appendix: Generalized spectra (contn'd.). Canonical decomposition 456 CHAPTER XIV. Generalized Spectra in the Presence of Approximate Identities. Representation Theory 1. Topological algebras with approximate identities. Representation theory Elementary measures of representations 474 CHAPTER XV. Topological Algebras with Involution. Representation Theory (contn'd.) 1. Preliminaries Certain particular (commutative) topological *-algebras and their representations SNAG Theorem (the classical case) Representations of generalized group algebras. SNAG Theorem (extended form) Abstract forms of "SNAG Theorem" type Appendix: Enveloping locally m-convex C*-algebras 497 BIBLIOGRAPHY 503 List of Symbols 527 INDEX 531