Abstract
We give sufficient conditions that allow contractible (resp., reflexive amenable) Banach algebras to be finite-dimensional and semisimple algebras. Moreover, we show that any contractible (resp., reflexive amenable) Banach algebra in which every maximal left ideal has a Banach space complement is indeed a direct sum of finitely many full matrix algebras. Finally, we characterize Hermitian*-algebras that are contractible.
Highlights
The purpose of this note is to establish the structure of some class of amenable Banach algebras
The admissible exact sequence splits if the right inverse of β is Banach left, bi- module, equivalently, ker β is a Banach space complement in ᐅ which is an Ꮽ-submodule
We present characterizations of contractible Banach algebras
Summary
The purpose of this note is to establish the structure of some class of amenable Banach algebras. The admissible exact sequence splits if the right inverse of β is Banach left, bi- module, equivalently, ker β is a Banach space complement in ᐅ which is an Ꮽ-submodule. Is every reflexive amenable Banach algebra finite-dimensional and semisimple? We will present two situations in which a contractible Banach algebra is finite-dimensional.
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