The Bochner-Schoenberg-Eberlein property for commutative Banach algebras, especially Fourier and Fourier-Stieltjes algebras

The classical Bochner-Schoenberg-Eberlein theorem characterizes the continuous functions on the dual group of a locally compact abelian group G G which arise as Fourier-Stieltjes transforms of elements of the measure algebra M ( G ) M(G) of G G . This has led to the study of the algebra of BSE-functions on the spectrum of an arbitrary commutative Banach algebra and of the concept of a BSE-algebra as introduced by Takahasi and Hatori. Since then BSE-algebras have been studied by several authors. In this paper we investigate BSE-algebras in the general context on the one hand and, on the other hand, we specialize to Fourier and Fourier-Stieltjes algebras of locally compact groups.