Non-Associative Normed Algebras: Volume 1, The Vidav-Palmer and Gelfand-Naimark Theorems
This first systematic account of the basic theory of normed algebras, without assuming associativity, includes many new and unpublished results and is sure to become a central resource for researchers and graduate students in the field. This first volume focuses on the non-associative generalizations of (associative) C*-algebras provided by the so-called non-associative Gelfand-Naimark and Vidav-Palmer theorems, which give rise to alternative C*-algebras and non-commutative JB*-algebras, respectively. The relationship between non-commutative JB*-algebras and JB*-triples is also fully discussed. The second volume covers Zel'manov's celebrated work in Jordan theory to derive classification theorems for non-commutative JB*-algebras and JB*-triples, as well as other topics. The book interweaves pure algebra, geometry of normed spaces, and complex analysis, and includes a wealth of historical comments, background material, examples and exercises. The authors also provide an extensive bibliography.
- Electronic book text
- 30 Jul 2014
- CAMBRIDGE UNIVERSITY PRESS
- Cambridge University Press (Virtual Publishing)
- Cambridge, United Kingdom
- 3 b/w illus. 20 exercises
Table of contents
Preface; 1. Foundations; 2. Beginning the proof of the non-associative Vidav-Palmer theorem; 3. Concluding the proof of the non-associative Vidav-Palmer theorem; 4. Jordan spectral theory; References; Symbol index; Subject index.
'... an impressive piece of work written by two experts in the theory of normed algebras whose multiplication may not be associative ... This review is certainly too brief to do justice to the rich contents of this book, which is a welcome addition to the literature on functional analysis and is definitely useful to anyone with an interest in this area and related topics, ranging from beginner operator theorists to people interested in specific problems involving non-associative algebras.' Daniel Beltita, Mathematical Reviews 'I dare say that this book is the first detailed treatise whose primary goal is the study of normed nonassociative algebras. The two authors are recognized experts in this topic, which combines functional analysis, geometry of Banach spaces and what could be called structure theory of algebras which are nearly associative.' Antonio Fernandez Lopez, zbMATH (www.zbmath.org)
About Miguel Cabrera García
Miguel Cabrera Garcia is Professor in the Department of Mathematical Analysis at the University of Granada, Spain. Angel Rodriguez Palacios is Professor in the Department of Mathematical Analysis at the University of Granada, Spain.