THE BANACH ALGEBRA ${\cal F}(S,T)$ AND ITS AMENABILITY OF COMMUTATIVE FOUNDATION $*$-SEMIGROUPS $S$ AND $T$

Abstract

In the present paper we shall first introduce the notion of the algebra ${\cal F}(S,T)$ 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 ${\cal F}(S,T)$ 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 ${\cal F}(S,T)$ is amenable and $K\big(\sigma (\overline{{\cal F}(S,T)})\big)$ is a compact topological group, where $\sigma (\overline {{\cal F}(S,T)})$ denotes the spectrum of the commutative Banach algebra $\overline{{\cal F}(S,T)}$ as a closed subalgebra of wap $(S \times T)$, the Banach algebra of weakly almost periodic continuous functions on $S \times T$.