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Jordan homomorphisms: A survey

Matej Brešar, Efim Zelmanov

TL;DR

This survey traces the historical development and current state of Jordan homomorphisms, focusing on the central question of expressing Jordan maps as sums of homomorphisms and antihomomorphisms. It highlights two key realizations: most Jordan algebras are special, realized inside $A^{(+)}$ or $H(A,*)$, and Jordan maps arise naturally in linear preserver problems, often forcing a decomposition into standard components. The article surveys foundational results for $A^{(+)}$ and $H(A,*)$, nonstandard examples, and modern approaches via commutator ideals, tetrad-eating T-ideals, and functional identities, culminating in related topics on Jordan derivations and Lie homomorphisms through the TKK framework. Together, these themes illuminate the extent to which Jordan homomorphisms are governed by underlying associative structures and how their behavior on key ideals determines a decomposition into homomorphisms and antihomomorphisms. The work underscores the interplay between Jordan and associative algebra techniques and their broad applicability across algebra and functional-analytic contexts.

Abstract

The paper surveys the history and state-of-the-art of the study of Jordan homomorphisms.

Jordan homomorphisms: A survey

TL;DR

This survey traces the historical development and current state of Jordan homomorphisms, focusing on the central question of expressing Jordan maps as sums of homomorphisms and antihomomorphisms. It highlights two key realizations: most Jordan algebras are special, realized inside or , and Jordan maps arise naturally in linear preserver problems, often forcing a decomposition into standard components. The article surveys foundational results for and , nonstandard examples, and modern approaches via commutator ideals, tetrad-eating T-ideals, and functional identities, culminating in related topics on Jordan derivations and Lie homomorphisms through the TKK framework. Together, these themes illuminate the extent to which Jordan homomorphisms are governed by underlying associative structures and how their behavior on key ideals determines a decomposition into homomorphisms and antihomomorphisms. The work underscores the interplay between Jordan and associative algebra techniques and their broad applicability across algebra and functional-analytic contexts.

Abstract

The paper surveys the history and state-of-the-art of the study of Jordan homomorphisms.
Paper Structure (12 sections, 24 theorems, 44 equations)

This paper contains 12 sections, 24 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. First motivation: Jordan homomorphisms are constituent parts of the Jordan algebra theory
  3. Second motivation: Jordan homomorphisms appear across mathematics
  4. Elementary observations
  5. Jordan homomorphisms on $A^{(+)}$: The early development
  6. Jordan homomorphisms on $H(A,\,*\,)$: The early development
  7. Examples of nonstandard Jordan homomorphisms
  8. Jordan homomorphisms on commutator ideals
  9. The approach via tetrad-eating T-ideals
  10. The approach via functional identities
  11. Jordan derivations
  12. Lie homomorphisms

Key Result

Theorem 3.1

A unital linear map $\varphi:M_n(\mathbb{C})\to M_n(\mathbb{C})$ is determinant preserving (i.e., $\det (\varphi(a))=\det(a)$ for every $a\in M_n(\mathbb{C})$) if and only if $\varphi$ is a Jordan automorphism.

Theorems & Definitions (40)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Remark 2.5
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Proposition 4.1
  • ...and 30 more