Figa-Talamanca–Herz algebras for restricted inverse semigroups and Clifford semigroups

Abstract

We develop the Figa-Talamanca–Herz algebras and the space of p-pseudomeasures to inverse semigroups with restricted semigroup algebras. Let 1<p,q<∞ be such that 1p+1q=1. We define the Banach algebra of p-pseudomeasures PMp(S) and the Figa-Talamanca–Herz algebras Aq(S). Then we show that Aq(S)∗=PMp(S). We characterize PMp(S) and Aq(S) for a Clifford semigroup, in the sense of p-pseudomeasures and Figa-Talamanca–Herz algebras of maximal subgroups of S, respectively. We also show that the character space of Aq(S) is equal to S for a Clifford semigroup S. As an example of these Banach algebras and restricted semigroup algebras, we discuss uniformly locally finite inverse semigroups.