Related papers: Inverse semigroups and the Cuntz-Li algebras
To each discrete left cancellative semigroup $S$ one may associate a certain inverse semigroup $I_l(S)$, often called the left inverse hull of $S$. We show how the full and the reduced C*-algebras of $I_l(S)$ are related to the full and…
We develop a theory of graph C*-algebras using path groupoids and inverse semigroups. Row finiteness is not assumed so that the theory applies to graphs for which there are vertices emitting a countably infinite set of edges. We show that…
We show that semigroup C*-algebras are groupoid C*-algebras.
Graph inverse semigroups generalize the polycyclic inverse monoids and play an important role in the theory of C*-algebras. This paper has two main goals: first, to provide an abstract characterization of graph inverse semigroups; and…
A semigroupoid is a set equipped with a partially defined associative operation. Given a semigroupoid \Lambda we construct a C*-algebra C*(\Lambda) from it. We then present two main examples of semigroupoids, namely the Markov semigroupoid…
We realize Kellendonk'?s C*-algebra of an aperiodic tiling as the tight C*-algebra of the inverse semigroup associated to the tiling, thus providing further evidence that the tight C*-algebra is a good candidate to be the natural…
We develop a structural theory of chirality for inverse semigroups and show how it propagates canonically to \'{e}tale groupoids and twisted groupoid $C^*$-algebras. Starting from inverse semigroup data equipped with admissible twist…
We develop a theory of type semigroups for arbitrary twisted, not necessarily Hausdorff \'etale groupoids. The type semigroup is a dynamical version of the Cuntz semigroup. We relate it to traces, ideals, pure infiniteness, and stable…
Cuntz and Li have defined a C*-algebra associated to any integral domain, using generators and relations, and proved that it is simple and purely infinite and that it is stably isomorphic to a crossed product of a commutative C*-algebra. We…
We investigate recent uniqueness theorems for reduced $C^*$-algebras of Hausdorff \'{e}tale groupoids in the context of inverse semigroups. In many cases the distinguished subalgebra is closely related to the structure of the inverse…
In a recent paper, Pardo and the first named author introduced a class of C*-algebras which which are constructed from an action of a group on a graph. This class was shown to include many C*-algebras of interest, including all Kirchberg…
In this paper, we prove that $C(SO_q(4)/SO_q(2))$ is isomorphic to the $C^*$-algebra of the tight groupoid $\mathcal{G}_{\mathrm{tight}}$ associated with the inverse semigroup generated by the standard generators of its classical limit…
In this article we survey some of the recent goings-on in the classification programme of C$^*$-algebras, following the interesting link found between the Cuntz semigroup and the classical Elliott invariant and the fact that the Elliott…
In this paper we study the structure of the $C^*$-algebra, generated by the representation of the paths semigroup on a partially ordered set (poset) and get the net of isomorphic $C^*$-algebras over this poset. We construct the extensions…
We prove that twisted groupoid C*-algebras are characterised, up to isomorphism, by having Cartan semigroups, a natural generalisation of normaliser semigroups of Cartan subalgebras. This extends the classic Kumjian-Renault theory to…
For any ring $R$, we introduce an invariant in the form of a partially ordered abelian semigroup $\mathrm{S}(R)$ built from an equivalence relation on the class of countably generated projective modules. We call $\mathrm{S}(R)$ the Cuntz…
We introduce a bivariant version of the Cuntz semigroup as equivalence classes of order zero maps generalizing the ordinary Cuntz semigroup. The theory has many properties formally analogous to KK-theory including a composition product. We…
We calculate the Cuntz semigroup of the tensor product A with A. We restrict our attention to C*-algebras A which are unital, simple, nuclear, stably finite, have stable rank one, absorbs the Jiang-Su algebra tensorially and satisfy the…
An amalgam of inverse semigroups [S,T,U] is full if U contains all of the idempotents of S and T. We show that for a full amalgam [S,T,U], the C*-algebra of the inverse semigroup amaglam of S and T over U is the C*-algebraic amalgam of…
We investigate the Banach Lie groupoids and inverse semigroups naturally associated to W*-algebras. We also present statements describing relationship between these groupoids and the Banach Poisson geometry which follows in the canonical…