This book provides a concise survey of the theory of zero product-determined algebras, which has been developed over the last 15 years. It is divided into three parts. The first part presents the purely algebraic branch of the theory, the second part presents the functional analytic branch, and the third part discusses various applications.
The book is intended for researchers and graduate students in ring theory, Banach algebra theory, and nonassociative algebra.
Author(s): Matej Brešar
Edition: 1
Publisher: Birkhäuser
Year: 2021
Language: English
Pages: 193
Tags: zero product-determined
Preface
Contents
Part I Algebraic Theory
1 Zero Product Determined Nonassociative Algebras
1.1 The Definition of a zpd Nonassociative Algebra
1.2 Symmetrically zpd Nonassociative Algebras
1.3 Stability Under Algebraic Constructions
2 Zero Product Determined Rings and Algebras
2.1 zpd Rings
2.2 Examples and Non-examples of zpd Algebras
2.3 The Finite-Dimensional Case
3 Zero Lie/Jordan Product Determined Algebras
3.1 Lie Algebras and Jordan Algebras
3.2 zLpd Algebras
3.3 zJpd Algebras
Part II Analytic Theory
4 Zero Product Determined Nonassociative Banach Algebras
4.1 Characters and the Limitations of the Algebraic Approach
4.2 The Definition of a zpd Nonassociative Banach Algebra
4.3 Point Derivations
5 Zero Product Determined Banach Algebras
5.1 Property B
5.2 Stability Under Analytic Constructions
5.3 Examples and Non-examples of zpd Banach Algebras
6 Zero Lie/Jordan Product Determined Banach Algebras
6.1 The Condition xy=yx=0
6.2 zLpd Banach Algebras
6.3 zJpd Banach Algebras
Part III Applications
7 Homomorphisms and Related Maps
7.1 Zero Product Preserving Maps
7.2 Commutativity Preserving Maps
7.3 Jordan Homomorphisms
8 Derivations and Related Maps
8.1 Characterizing Derivations by Action on Zero Products
8.2 Local Derivations
8.3 Derivations and Quasinilpotent Elements
9 Miscellany
9.1 Commutators and Special-Type Elements
9.2 Orthogonality Conditions on n-Linear Maps
9.3 Nonassociative Products of Matrices
References
Index
