Description
Book SynopsisThis 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.
Trade Review“This book deals with the theory of locally convex algebras, in general, and of generalized
B_-algebras (GB_-algebras in short) in particular. It is well written and self-contained.” (Lahbib Oubbi, Mathematical Reviews, November, 2023)
“The book has been written by specialists that are actively working in the field. The choice of the presented material has been done with great care. The bibliography contains all classical monographs, all important papers, and most recent ones. The book leads the reader 'smoothly' ... . It will therefore serve as an excellent introduction to this theory for graduate students. It should also provide a valuable reference source for researchers in the field.” (Andrzej Sołtysiak, zbMATH 1498.46001, 2022)
Table of Contents1. Introduction.- 2. A Spectral Theory for Locally Convex Algebras.- 3. Generalized B*-Algebras: Functional Representation Theory.- 4. Commutative Generalized B*-Algebras: Functional Calculus and Equivalent Topologies.- 5. Extended C*-Algebras and Extended W*-Algebras.- 6. Generalized B*-Algebras: Unbounded *-Representation Theory.- 7. Applications I: Miscellanea.- 8. Applications II: Tensor Products.
