Actions of inverse semigroups arising from partial actions of groups

Abstract

In this work we present a definition of crossed product for actions of inverse semigroups on C * -algebras, without resorting to covariant representations as done by Sieben in related work. We also show the existence of an isomorphism between the crossed product by a partial action of a group G and the crossed product by a related action of S(G), an inverse semigroup associated to G introduced by the first named author.

Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics

“…If we assume that A is associative, which we will, then using the assumption that A and each D s , s ∈ S, have local units 1 one can show that L is an associative ring (see [22,Theorem 3.4]). (2) Then, we consider the ideal N = aδ r − aδ s | r, s ∈ S, r ≤ s and a ∈ D r , i.e.…”

“…This class of rings was introduced by Exel and Vieira (see e.g. [4,22]) and generalizes the class of partial skew group rings (see [22,Theorem 3.7]). Our interest to study this class of rings comes from its connections with topological dynamics (see Section 4), and the fact that any Steinberg algebra, associated with a Hausdorff and ample groupoid, can be realized as a skew inverse semigroup ring (see [4]).…”

An inverse semigroup approach to the C*-algebras and crossed products of cancellative semigroups

“…Corollary 5.9. Using the results of [21], one has the following commutative diagram where each map is a surjective *-homomorphism: 10. Note that the last arrow is reversed due to the fact that C * r (S) is generated by a subrepresentation of the left regular representation of S * (or I l (S)).…”

“…In [10] it was shown, that the partial actions and partial representations of a group G are in one-to-one correspondence with actions and representations of a special inverse semigroup S(G). Moreover, an isomorphism between a partial crossed product by G and a crossed product by S(G) was proved in [10] and earlier in [30]. We recall these results.…”

“…Following [10] S(G) is a semigroup generated by elements t g for all g ∈ G satisfying the following relations:…”