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

Abstract

a b s t r a c tWe 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 1 p + 1 q = 1. We define the Banach algebra of p-pseudomeasures PM p (S) and the FigaTalamanca-Herz algebras A q (S). Then we show that A q (S) * = PM p (S). We characterize PM p (S) and A q (S) for a Clifford semigroup, in the sense of p-pseudomeasures and FigaTalamanca-Herz algebras of maximal subgroups of S, respectively. We also show that the character space of A q (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.