© Hindawi Publishing Corp. THE STRUCTURE OF A SUBCLASS OF AMENABLE BANACH ALGEBRAS

We give sufficient conditions that allow contractible (resp., reflexive amenable) Banach al-gebras to be finite-dimensional and semisimple algebras. Moreover, we show that any con-tractible (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. 2000 Mathematics Subject Classification: 46H25, 18G55, 46M20. 1. Introduction. The purpose of this note is to establish the structure of some class of amenable Banach algebras. Let be a Banach algebra over the complex field C. We define a Banach left -module to be a Banach space which is also a unital left -module such that the linear map × → , (a,x) → ax, is continuous. Right mod-ules are defined analogously. A Banach -bimodule is a Banach space with a structura