Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Derivations and homomorphisms in commutator-simple algebras

J. Alaminos, M. Brešar, J. Extremera, M. L. C. Godoy, A. R. Villena

Abstract

We call an algebra $A$ commutator-simple if $[A,A]$ does not contain nonzero ideals of $A$. After providing several examples, we show that in these algebras derivations are determined by a condition that is applicable to the study of local derivations. This enables us to prove that every continuous local derivation $D\colon L^1(G)\to L^1(G)$, where $G$ is a unimodular locally compact group, is a derivation. We also give some remarks on homomorphism-like maps in commutator-simple algebras.

Derivations and homomorphisms in commutator-simple algebras

Abstract

We call an algebra commutator-simple if does not contain nonzero ideals of . After providing several examples, we show that in these algebras derivations are determined by a condition that is applicable to the study of local derivations. This enables us to prove that every continuous local derivation , where is a unimodular locally compact group, is a derivation. We also give some remarks on homomorphism-like maps in commutator-simple algebras.
Paper Structure (4 sections, 22 theorems, 56 equations)

This paper contains 4 sections, 22 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Commutator-simple algebras
  3. Derivations
  4. Jordan homomorphisms

Key Result

Proposition 2.1

Every commutative algebra is commutator-simple.

Theorems & Definitions (41)

  • Definition 1.1
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 31 more