Group Duality and Isomorphisms of Fourier and Fourier-stieltjes Algebras from a W*-algebra Point of View

0. Introduction. In this paper two commutative, semisimple Banach algebras with involution, viz., A(G) and B(G), (called the Fourier algebra of G and the Fourier-Stieltjes algebra of G, respectively) are defined (as in [5]) for an arbitrary locally compact topological group G. [Note. As algebras A(G) and B(G) are subalgebras of the algebra of bounded continuous functions on G.] The main purpose of this paper is to show that these algebras deeply reflect the structure of G even though they apparently are "trivial" in the sense that multiplication is defined pointwise-not by convolution as, for example, in the case of L(G). The first main result is closely related to a well-known duality theorem, cf. [4], [5], [9], [lO], [13], and it has a relatively short proof. The group G is recovered topologically as a special subset of the spectrum of B(G) (or as the spectrum of A(G)), and by virtue of the fact that the dual of B(G) or of A(G)y denoted B(G)', A(G)' respectively, can be naturally equipped with an algebraic structure, G can be recovered algebraically as well. The main difference between our Theorem 1 and the above results is that we proceed directly to recovering the group G in the 5(G), B{G)' context and deduce the duality theorem in the A(G), A(G)' context as an easy corollary; whereas the other papers attack the A(G), A(G) duality first and, except for [4] and a long complicated formulation in [13], do not discuss the process of recovering the group from the B(G), B{G) duality. We also obtain the formula in Theorem 1 (i) and further clarify the relationship between B(G) and the representation theory of G.