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

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 1 p + 1 q = 1 . We define the Banach algebra of p -pseudomeasures PM p ( S ) and the Figa-Talamanca–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 Figa-Talamanca–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.