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. This is the first in a series of papers in which we have investigated a similar relation on inverse semigroups. We use a new concept of "restricted" representations and study the restricted semigroup algebras and corresponding C * -algebras.
In this paper, we characterize pseudo-contractibility of 1 (S), where S is a uniformly locally finite inverse semigroup. As a consequence, we show that for a Brandt semigroup S = M 0 (G, I), the semigroup algebra 1 (S) is pseudo-contractible if and only if G and I are finite. Moreover, we study the notions of pseudo-amenability and pseudo-contractibility of a semigroup algebra 1 (S) in terms of the amenability of S.
Mathematics Subject Classification (2000). Primary 43A20, 20M18; Secondary 16E40.
We investigate the algebraic structure of the spectrum Ω of L∞(G) for a locally compact group G. In contrast to the compact and discrete cases, when G has neither of these properties, Ω is never a semigroup. For σcompact G we determine exactly when the product of two elements of Ω. is in Ω, but we present an example which suggests that for general groups the underlying set theory may have an effect. Our principal tool, which has independent interest, is a topological structure theorem for the LB‐compactification of an arbitrary locally compact group.
We introduce the notion of the Fourier and Fouier-Stieltjes algebra of a topological * -semigroup and show that these are commutative Banach algebras. For a class of foundation semigroups, we show that these are preduals of von Neumann algebras.
We show that for a compact hypergroup K, the hypergroup algebra L1(K) is amenable as a Banach algebra if the set of hyperdimensions of irreducible representations of K is bounded above. Conversely if L1(K) is amenable, the set of ratios of the hyperdimension to the dimension of irreducible representations of K is bounded above. These are equivalent for compact commutative hypergroups.
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