Derivations and homomorphisms in commutator-simple algebras. (English) Zbl 1533.43003
Summary: 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 \(D:L^1(G)\to L^1(G)\), where \(G\) is a unimodular locally compact group, is a derivation. We also give some remarks on homomorphism-like maps in commutator-simple algebras.
MSC:
| 43A20 | \(L^1\)-algebras on groups, semigroups, etc. |
| 47B47 | Commutators, derivations, elementary operators, etc. |
| 43A22 | Homomorphisms and multipliers of function spaces on groups, semigroups, etc. |
| 46H20 | Structure, classification of topological algebras |
| 47L10 | Algebras of operators on Banach spaces and other topological linear spaces |
| 16W20 | Automorphisms and endomorphisms |
| 16W25 | Derivations, actions of Lie algebras |
Keywords:
derivation; automorphism; antiautomorphism; Jordan automorphism; local derivation; local automorphism; group algebraReferences:
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