The banach algebra F(S, T) and its amenability of commutative foundation *-semigroups S and T

Abstract

In the present paper we shall first introduce the notion of the algebra F(S, T) of two topological *-semigroups S and T in terms of bounded and weakly continuous *-representations of S and T on Hilbert spaces. In the case where both S and T are commutative foundation *-semigroups with identities it is shown that F(S, T) is identical to the algebra of the Fourier transforms of bimeasures in BM(S*, T*), where S* (T*, respectively) denotes the locally compact Hausdorff space of all bounded and continuous *-semicharacters on S(T, respectively) endowed with the compact open topology. This result has enabled us to make the bimeasure Banach space BM(S*, T*) into a Banach algebra. It is also shown that the Banach algebra F(S, T) is amenable and is a compact topological group, where denotes the spectrum of the commutative Banach algebra as a closed subalgebra of wap (S ×T), the Banach algebra of weakly almost periodic continuous functions on S × T.