Topological Algebras with Involution (eBook)

Cover 1
Contents 7
Table of Contents 7
Introduction 11
General Theory 19
Background material 21
Topological algebras. Definitions, examples 21
Arens-Michael algebras. Basic examples 30
Topological *-algebras 44
Basic properties 49
Examples of topological algebras with continuous or non-continuous involution 52
Unitization of a topological (*-)algebra 56
Quotient of a topological (*-)algebra 58
Local and global spectrum 60
Non-Banach Arens-Michael algebras 77
Square roots in Arens-Michael algebras 81
Q and advertibly complete algebras. Basic properties, examples 87
Locally C*-algebras 117
Definitions. Examples 119
Basic properties of locally C*-algebras 128
Gel'fand-Naimark type theorem 131
Commutative Gel'fand-Naimark type theorems 133
Functional Calculus 137
The cone of positive elements 143
The pre-C*-algebras of the Arens-Michael decomposition of a locally C*-algebra are (automatically) complete 148
Approximate identities 153
Quotient of a locally C*-algebra 158
Representation theory 161
Positive linear forms and extreme points 162
*-Representations 178
The GNS-construction 183
Represent ability of linear forms 192
On the existence of positive linear forms and ^representations 198
Continuity of positive linear forms 203
Continuity of the involution 210
Continuity of ^-representations and positive linear forms, independent of continuity of the involution 214
Structure space of an m*—convex algebra 225
Enveloping locally C*-algebra of an m*-convex algebra 226
m*-Convex algebras with C*-enveloping algebra 239
Structure space of a locally C*-algebra 254
Existence of faithful ^representations 262
Spectrally bounded algebras 269
Structure theorems on topological (*-)algebras 276
Hermitian and symmetric topological *—algebras 283
Around hermiticity and symmetry 285
Examples of hermitian and symmetric topological *- algebras 294
Ptak function and Ptak inequality 297
The Shirali-Ford theorem 314
Symmetry via positive linear forms and ^-representations 320
Existence of faithful ^representations 330
Applications 335
Integral representations. Uniqueness of topology 337
The Riesz representation theorem 337
Positive measures 341
Integral representations of linear forms 343
Classical Bochner-Weil-Raikov theorems 346
Integral representations of linear forms in terms of a technique of G. Lumer 350
Bochner type integral representations 357
The abstract Bochner-Weil-Raikov theorem 362
Uniqueness of topology on semisimple m-convex algebras 366
Applications to function algebras from distribution theory 372
An application concerning involution 376
Tensor products of topological *-algebras 381
Preliminaries. Definitions 382
Positive linear forms and ^-representations 392
Factor and type / *-representations 397
Tensor products of locally C*-algebras 404
Inverse limit preserving tensorial topologies 406
The spectrum of an elementary tensor 420
The enveloping locally C*-algebra of tensor products of topological *-algebras 427
The C*-enveloping algebra of tensor products of Banach *-algebras 427
The enveloping locally C*-algebra of tensor products of m*-convex algebras 430
The structure space of a tensor product topological *-algebra 436
Indecomposable positive linear forms 437
The structure space of a tensor product Banach *- algebra 445
The structure space of a tensor product Frechet *- algebra 448
Properties of Permanence 457
*-Semisimplicity 457
Hermiticity and symmetry 465
Bibliography 469
Index 505