THE BANACH ALGEBRA ℱ(S, T) AND ITS AMENABILITY OF COMMUTATIVE FOUNDATION ∗-SEMIGROUPSSANDT

In the present paper we shall first introduce the notion of the algebra ℱ(S, T) of two topological ∗-semigroupsSandTin terms of bounded and weakly continuous ∗-representations ofSandTon Hilbert spaces. In the case where bothSandTare commutative foundation ∗-semigroups with identities it is shown that ℱ(S, T) is identical to the algebra of the Fourier transforms of bimeasures inBM(S * , T *), whereS *(T *, respectively) denotes the locally compact Hausdorff space of all bounded and continuous ∗-semicharacters onS(T, respectively) endowed with the compact open topology. This result has enabled us to make the bimeasure Banach spaceBM(S*, T*) into a Banach algebra. It is also shown that the Banach algebra ℱ(S, T) is amenable and K ( σ ( ℱ ( S , T ) ¯ ) ) is a compact topological group, where σ ( ℱ ( S , T ) ¯ ) denotes the spectrum of the commutative Banach algebra ℱ ( S , T ) ¯ as a closed subalgebra of wap (S×T). the Banach algebra of weakly almost periodic continuous functions onS×T. 2010Mathematics Subject Classification: 43A65, 22A25, 43A35, 43A10. Key words and phrases: Topological semigroup, Representation, Bimeasure, Banach algebra, Fourier-Stieltjes algebra.