Abstract
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.
Highlights
Introduction and PreliminariesAn inverse semigroup S is a discrete semigroup such that, for each s ∈ S, there is a unique element s∗ ∈ S such that ss∗s s, s∗ss∗ s∗.One can show that s → s∗ is an involution on S 1
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
The left regular representation λ of an inverse semigroup loses its connection with positive definite functions
Summary
An inverse semigroup S is a discrete semigroup such that, for each s ∈ S, there is a unique element s∗ ∈ S such that ss∗s s, s∗ss∗ s∗. 1.4 for x ∈ S and ξ, η ∈ 2 S fails in general This makes it difficult to study positive definite functions 3 and Fourier Fourier-Stieltjes algebras on semigroups 4. In the new convolution product, positive definite functions fit naturally with a restricted version of the left regular representation λr. The main objective of 5 is to change the convolution product on the semigroup algebra to restore the relation between positive definite functions and left regular representation 6. Banach algebra in the sense of 7 , where the order structure comes from the set of extendable restricted positive definite functions as the positive cone. For each inverse semigroup S, the states on the ∗-algebra CS the vector space over S spanned by S with convolution and involution comes from S are defined by Milan in 8. The amenability results for Fourier and Fourier-Stieltjes algebras on groups are surveyed in 10
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