Inverse semigroups and combinatorial C*-algebras

Abstract

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.

Diagonal-preserving graded isomorphisms of Steinberg algebras

“…, where G θ is the groupoid constructed in [1] from a partial action θ of a discrete group G on a totally disconnect locally compact Hausdorff space X , and c φ : G θ → Γ is the cocycle induced by a group homomorphism φ : G → Γ such that the group-ring R(ker φ ) has no zero-divisors and only trivial units; • the pair (G tight (S ), c φ ), where G tight (S ) is the tight groupoid of an E * -unitary inverse semigroup (see [23,24]), and c φ is the cocycle induced by a homomorphism φ : S → Γ such that there is a dense subset X ⊆Ê tight such that the groupring R(G x ), where G x is the group {[s, x] ∈ G tight (S ) : φ (s) = e, θ s (x) = x}, has no zero-divisors and only trivial units for all x ∈ X .…”

Inverse semigroup actions as groupoid actions

“…We begin with the following well-known result (see [4] and also [6]): (1) , that is, S = G (1) , and that for all s, t ∈ S and g ∈ s ∩ t, there is r ∈ S with g ∈ r ⊆ s ∩ t.…”

“…With a different, Hausdorff topology, this action of S has already been studied in [4,8]. Our non-Hausdorff topology has the following crucial feature: if Gr(S) is the étale groupoid of germs for the action of S onÊ, then the category of actions of S on topological spaces is equivalent to the category of actions of Gr(S) on topological spaces.…”

C⁎-algebras associated to Boolean dynamical systems

“…If B and L are countable then, since G tight (T ) is the tight groupoid of an countable * -inverse semigroup, G tight (T ) is anétale, second countable, topological groupoid [13]. We know that G tight (T ) is Hausdorff and amenable.…”