Derivations and homomorphisms in commutator-simple algebras

Abstract

We call an algebra A A commutator-simple if [ A , A ] [A,A] does not contain nonzero ideals of A 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 : L 1 ( G ) → L 1 ( G ) D\colon L^1(G)\to L^1(G) , where G G is a unimodular locally compact group, is a derivation. We also give some remarks on homomorphism-like maps in commutator-simple algebras.