Topological Algebras with Involution (eBook)

M. Fragoulopoulou (Autor)

eBook Download: PDF

2005 | 1. Auflage
512 Seiten
Elsevier Science (Verlag)
978-0-08-046122-9 (ISBN)

Infos

This book familiarizes both popular and fundamental notions and techniques from the theory of non-normed topological algebras with involution, demonstrating with examples and basic results the necessity of this perspective. The main body of the book is focussed on the Hilbert-space (bounded) representation theory of topological *-algebras and their topological tensor products, since in our physical world, apart from the majority of the existing unbounded operators, we often meet operators that are forced to be bounded, like in the case of symmetric *-algebras. So, one gets an account of how things behave, when the mathematical structures are far from being algebras endowed with a complete or non-complete algebra norm. In problems related with mathematical physics, such instances are, indeed, quite common.

Key features:

- Lucid presentation
- Smooth in reading
- Informative
- Illustrated by examples
- Familiarizes the reader with the non-normed *-world
- Encourages the hesitant
- Welcomes new comers.

- Well written and lucid presentation.
- Informative and illustrated by examples.
- Familiarizes the reader with the non-normed *-world.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 26.7.2005
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Technik
• ISBN-10: 0-08-046122-0 / 0080461220
• ISBN-13: 978-0-08-046122-9 / 9780080461229