For a unital foundation topological *-semi group S whose representations separate points of S, we show that the spectrum of the Fourier-Stieltjes algebra B(S) is a compact semi topological semi group. We also calculate B(S) for several examples of S.

In the present paper for a large family of topological semigroups, namely foundation semigroups, for which topological groups and discrete semigroups are elementary examples, it is shown that R(S) is the dual of a function algebra.

We introduce the notion of LMIP ( A), where A is a Banach space, I is an index set and 1 £ p <¥. We find necessary and sufficient conditions for which LMIP ( A) is a Banach algebra and investigate amenability of this Banach algebra. Applications to lP(S) (1 £ p <¥), where S is a Brandt semigroup, are also given.

In the present paper we give a partially negative answer to a conjecture of Ghahramani, Runde and Willis. We also discuss the derivation problem for both foundation semigroup algebras and Clifford semigroup algebras. In particular, we prove that if S is a topological Clifford semigroup for which Es is finite, then H1(M(S),M(S))={0}.

In this paper we define φ-Connes module amenability of a dual Banach algebra A where φ is a bounded wk∗-module homomorphism from A to A. We are mainly concerned with the study of φ-module normal virtual diagonals. We show that if S is a weakly cancellative inverse semigroup with subsemigroup E of idempotents, χ is a bounded wk∗-module homomorphism from l1(S) to l1(S) and l1(S) as a Banach module over l1(E) is χ-Connes module amenable, then it has a χ-module normal virtual diagonal. In the case χ =id, the converse holds

In this paper we define a congruence ~ on inverse semigroup S such that amenability of S is equivalent to amenability of S/ ~. We study module amenability of semigroup algebra i1(S/ ~) when S is an inverse semigroup with idempotents E and prove that it is equivalent to module amenability of i1 (S). The main difference of this action with the more studied trivial action is that in this case the corresponding homomorphic image is a Clifford semigroup rather than a discrete group.