C *-crossed products by partial actions and actions of inverse semigroups

Bull Braz Math Soc, New Series

2008

242

499

We describe a special class of representations of an inverse semigroup S on Hilbert's space which we term tight. These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the tight spectrum, which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way. These representations are moreover shown to correspond to representations of the C*-algebra of the groupoid of germs for the action of S on its tight spectrum. We then treat the case of certain inverse semigroups constructed from semigroupoids, generalizing and inspired by inverse semigroups constructed from ordinary and higher rank graphs. The tight representations of this inverse semigroup are in one-to-one correspondence with representations of the semigroupoid, and consequently the semigroupoid algebra is given a groupoid model. The groupoid which arises from this construction is shown to be the same as the boundary path groupoid of Farthing, Muhly and Yeend, at least in the singly aligned, sourceless case.

Section: R Exelmentioning

confidence: 99%

Inverse semigroups and combinatorial C*-algebras

Bull Braz Math Soc, New Series

2008

242

499

“…The crossed product by an action of an inverse semigroup was introduced by Nándor Sieben in [5] in 1994. In this definition, he used covariant representations of an action.…”

Section: Covariant Representationsmentioning

confidence: 99%

“…We will outline his definition and we will show that our definition is equivalent to his. See [5] for further details on Sieben's work. (i) ν s π(a)ν s * = π(β s (a)) for all a ∈ E s * (covariance condition), (ii) ν s has initial space span{π (E s * )H} and final space span{π (E s )H}.…”

Section: Covariant Representationsmentioning

confidence: 99%

Actions of inverse semigroups arising from partial actions of groups

Journal of Mathematical Analysis and Applications

Vieira

2010

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.

“…This problem was first considered in Sieben's [14] master thesis, where a partial answer was given in the case of actions on C * -algebras. The inverse subgroup constructed there, however, is not intrinsically obtained from G, but it depends also on the partial action under consideration.…”

Section: Introductionmentioning

confidence: 99%

Partial actions of groups and actions of inverse semigroups

1998

Proc. Amer. Math. Soc.

184

156

Abstract. Given a group G, we construct, in a canonical way, an inverse semigroup S(G) associated to G. The actions of S(G) are shown to be in one-to-one correspondence with the partial actions of G, both in the case of actions on a set, and that of actions as operators on a Hilbert space. In other words, G and S(G) have the same representation theory.We show that S(G) governs the subsemigroup of all closed linear subspaces of a G-graded C * -algebra, generated by the grading subspaces. In the special case of finite groups, the maximum number of such subspaces is computed.A "partial" version of the group C * -algebra of a discrete group is introduced. While the usual group C * -algebra of finite commutative groups forgets everything but the order of the group, we show that the partial group C * -algebra of the two commutative groups of order four, namely Z/4Z and Z/2Z ⊕ Z/2Z, are not isomorphic.