This book reviews the theory of 'generalized B*-algebras' (GB*-algebras), a class of complete locally convex *-algebras which includes all C*-algebras and some of their extensions. A functional calculus and a spectral theory for GB*-algebras is presented, together with results such as Ogasawara's commutativity condition, Gelfand–Naimark type theorems, a Vidav–Palmer type theorem, an unbounded representation theory, and miscellaneous applications. Numerous contributions to the subject have been made since its initiation by G.R. Allan in 1967, including the notable early work of his student P.G. Dixon. Providing an exposition of existing research in the field, the book aims to make this growing theory as familiar as possible to postgraduate students interested in functional analysis, (unbounded) operator theory and its relationship to mathematical physics. It also addresses researchers interested in extensions of the celebrated theory of C*-algebras.

1 Introduction 1

2 A Spectral Theory for Locally Convex Algebras 9

2.1 Basic Definitions and Notation 9

2.2 The Set of Bounded Elements. Radius of Boundedness 11

2.3 Spectrum and Spectral Radius 20

2.4 A Functional Calculus 30

2.5 The Carrier Space 32

3 Generalized B*-AIgebras: Functional Representation Theory 39

3.1 Hermitian and Symmetric Locally Convex *-Algcbras 39

3.2 Some Results on C*-Algebras 45

3.3 GB*-Algebras 51

3.4 Commutative GB*-Algebras: Functional Representation Theory 62

3.5 C*-Like Locally Convex *-Algebras as GB*-Algebras 70

4 Commutative Generalized B*-A!gebras: Functional Calculus and Equivalent Topologies 79

4.1 Functional Calculus 79

4.2 Positive Linear Functionals 83

4.3 Equivalent Topologies 91

4.4 A*-Algebra of Functions with no GB*-Topology 99

5 Extended C*-Algebras and Extended W*-Algebras 103

5.1 O*-Algebras and Unbounded *-Representalions 103

5.2 Uniform Topologies on O*-Algebras 106

5.3 Extended C*-AIgcbras 114

5.4 Left Extended W*-Algebras of Unbounded Hilbert Algebras 116

6 Generalized B*-Algebras: Unbounded *-Representation Theory 127

6.1 A Functional Calculus 127

6.2 Positive Elements 129

6.3 *-Representations of GB*-Algebras 137

6.4 Ogasawara's Commutativity Condition in GB*-Algebras 154

7 Applications I: Miscellanea 157

7.1 The Completion of a C*-Aigebra Under a Locally Convex Topology 157

7.2 Continuity of Positive Linear Functionals of GB*-Algebras 167

7.3 Every GB*-Algebra has a Bounded Approximate Identity 170

7.4 Inverse Limits and Quotients of GB*-Algebras 172

7.5 A Vidav-Palmer Theorem for GB*-Algebras 178

8 Applications II: Tensor Products 211

8.1 Prerequisites: Examples 211

8.2 B*-Collections in Tensor Product Locally Convex *-Algebras 215

8.3 Tensor Product GB*-Algebras 218

8.4 GB*-Nuclearily 223

8.5 Some Applications on Tensor Product GB*-Algebras 226

Bibliography 229

Index 235

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