Derivations and homomorphisms in commutator-simple algebras

TL;DR

This paper explores the structure of commutator-simple algebras, showing derivations are determined by specific conditions and proving that continuous local derivations on certain group algebras are actual derivations.

Contribution

It introduces the concept of commutator-simple algebras, characterizes derivations in these algebras, and proves that local derivations on $L^1(G)$ are derivations for unimodular groups.

Findings

Derivations in commutator-simple algebras are characterized by specific conditions.

Every continuous local derivation on $L^1(G)$ for unimodular groups is a derivation.

Provides examples and remarks on homomorphism-like maps in these algebras.

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.