Abstract:The relation between representations and positive definite functions is a key concept in harmonic analysis on topological groups. Recently this relation has been studied on topological groupoids. In this paper, we investigate the concept of restricted positive definite functions and their relation with restricted representations of an inverse semigroup. We also introduce the restricted Fourier and Fourier-Stieltjes algebras of an inverse semigroup and study their relation with the corresponding algebras on the… Show more
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.
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.
We discuss amenability of the restricted Fourier-Stieltjes algebras on inverse semigroups. We show that, for an E-unitary inverse semigroup, amenability of the restricted Fourier-Stieltjes algebra is related to the amenability of an associated Banach algebra on a Fell bundle.
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