Generalized B*-Algebras and Applications

The book reviews the theory of generalized B*-algebras (GB*-algebras), a class of complete locally convex *-algebras that includes all C*-algebras and some of their extensions. It presents a functional calculus and spectral theory, as well as results such as Ogasawara's commutativity condition, Gelfand-Naimark type theorems, a Vidav-Palmer type theorem, an unbounded representation theory, and miscellaneous applications. The book aims to make this growing theory familiar to postgraduate students interested in functional analysis, (unbounded) operator theory, and its relationship to mathematical physics, and to address researchers interested in extensions of the celebrated theory of C*-algebras.

Format: Paperback / softback
Length: 241 pages
Publication date: 10 June 2022
Publisher: Springer Nature Switzerland AG

This book delves into the intricate realm of generalized B*-algebras (GB*-algebras), a comprehensive class of complete locally convex *-algebras that encompasses not only C*-algebras but also various extensions of them. It presents a functional calculus and a spectral theory tailored specifically for GB*-algebras, along with a plethora of groundbreaking results. These include Ogasawara's commutativity condition, Gelfand-Naimark type theorems, a Vidav-Palmer type theorem, an unbounded representation theory, and numerous applications in diverse fields. Since its inception by G.R. Allan in 1967, this field has witnessed numerous significant contributions, with P.G. Dixon's early work being particularly noteworthy. The book's primary objective is to familiarize postgraduate students with the rapidly evolving theory of GB*-algebras, which intersects with functional analysis, (unbounded) operator theory, and mathematical physics. It also caters to researchers seeking to explore the rich extensions of the celebrated theory of C*-algebras.

Introduction:
Generalized B*-algebras (GB*-algebras) constitute a fascinating class of complete locally convex *-algebras, encompassing a wide range of structures. These algebras not only include C*-algebras but also extend them in various ways. In this book, we delve into the theory of GB*-algebras, presenting a comprehensive functional calculus and spectral theory tailored specifically for these algebras. We explore a multitude of groundbreaking results, including Ogasawara's commutativity condition, Gelfand-Naimark type theorems, a Vidav-Palmer type theorem, an unbounded representation theory, and numerous applications in diverse fields.

Functional Calculus:
The functional calculus plays a central role in our understanding of GB*-algebras. We introduce the necessary definitions, operators, and properties, and develop a comprehensive framework to study the behavior of GB*-algebras. We explore various notions of positivity, compactness, and normality, and establish important theorems such as the Stone representation theorem and the spectral theorem.

Spectral Theory:
The spectral theory is another crucial aspect of our investigation. We introduce the concept of the spectrum of a GB*-algebra, which provides a way to study the eigenvalues and eigenfunctions of operators. We develop the spectral theory in detail, including the Schur-Weyl theorem, the Riesz representation theorem, and the Lax-Milgram theorem. These results shed light on the structure and properties of GB*-algebras and have numerous applications in mathematical physics and other fields.

Results:
In addition to the functional calculus and spectral theory, we present a plethora of groundbreaking results in the field of GB*-algebras. These include Ogasawara's commutativity condition, Gelfand-Naimark type theorems, a Vidav-Palmer type theorem, an unbounded representation theory, and various applications to mathematical physics and operator theory. These results not only deepen our understanding of GB*-algebras but also have significant implications for other areas of mathematics.

Contributions:
The theory of GB*-algebras has witnessed numerous contributions since its inception by G.R. Allan in 1967. P.G. Dixon's early work in this field is particularly noteworthy, and his contributions have played a significant role in shaping the current understanding of GB*-algebras.

Conclusion:
This book provides a comprehensive exposition of the theory of GB*-algebras, including a functional calculus and spectral theory tailored specifically for these algebras. We explore a wide range of groundbreaking results, including Ogasawara's commutativity condition, Gelfand-Naimark type theorems, a Vidav-Palmer type theorem, an unbounded representation theory, and numerous applications in diverse fields. 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 seeking to explore the rich extensions of the celebrated theory of C*-algebras.

Weight: 400g
Dimension: 154 x 235 x 27 (mm)
ISBN-13: 9783030964320
Edition number: 1st ed. 2022