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On an approximate automorphism on a \(C^{*}\)-algebra. (English) Zbl 1055.47032

In the first part of the paper, the author considers mappings on Banach *-algebras which are approximately linear, approximately multiplicative and approximately self-adjoint (see the paper for precise definitions). He presents results which say that for every mapping \(f\) having such “approximate” properties, there exists an algebra *-homomorphism which is close to \(f\). The obtained theorems are termed as generalized Hyers–Ulam–Rassias stability results concerning *-homomorphisms of Banach *-algebras. In the last section of the paper, the author investigates the stability of the automorphisms of unital \(C^*\)-algebras. He shows that under certain conditions, if a bijective multiplicative map on such an algebra is approximately linear and approximately self-adjoint in some sense, then it is automatically a *-automorphism.

MSC:

47B48 Linear operators on Banach algebras
46L40 Automorphisms of selfadjoint operator algebras
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