Non-Associative Normed Algebras

The first systematic account of the basic theory of normed algebras, without assuming associativity. Sure to become a central resource.

Miguel Cabrera García (Author), Ángel Rodríguez Palacios (Author)

9781107043114, Cambridge University Press

Hardback, published 12 April 2018

754 pages
24 x 16 x 4.6 cm, 1.28 kg

'Just as the first volume of this monograph, the second volume under review here deserves the attention of nearly everybody interested in functional analysis and its applications, as well as of many people with an interest in several complex variables or in the theory of Lie groups. The authors skillfully present in a self-contained and well-motivated manner such basic topics as differential calculus on Banach spaces, infinitedimensional holomorphy, vector fields and their flows, Banach-Lie group structure on the holomorphic automorphisms of a bounded domain in a Banach space, or joint spectral radius in Banach algebras. This book is a really great achievement on the part of the authors and studying it or even parts of it will be both pleasant and profitable to the reader.' Daniel Beltita, Mathematical Reviews

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first systematic account of the basic theory of normed algebras, without assuming associativity, includes many new and unpublished results and is sure to become a central resource for researchers and graduate students in the field.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 second volume revisits JB*-triples, covers Zel'manov's celebrated work in Jordan theory, proves the unit-free variant of the Vidav–Palmer theorem, and develops the representation theory of alternative C*-algebras and non-commutative JB*-algebras. This completes the work begun in the first volume, which introduced these algebras and discussed the so-called non-associative Gelfand–Naimark and Vidav–Palmer theorems. This book interweaves pure algebra, geometry of normed spaces, and infinite-dimensional complex analysis. Novel proofs are presented in complete detail at a level accessible to graduate students. The book contains a wealth of historical comments, background material, examples, and an extensive bibliography.

Preface
5. Non-commutative JBW*-algebras, JB*-triples revisited, and a unit-free Vidav–Palmer type non-associative theorem
6. Representation theory for non-commutative JB*-algebras and alternative C*-algebras
7. Zel'manov approach
8. Selected topics in the theory of non-associative normed algebras
Bibliography of Volume 1
Additional bibliography to Volume 2
Symbol index for Volume 1
Subject index for Volume 1
Symbol index for Volume 2
Subject index for Volume 2.

Subject Areas: Algebraic topology [PBPD], Functional analysis & transforms [PBKF], Algebra [PBF]