C*-algebras associated to Boolean dynamical systems

The goal of this talk is to present the C*-algebra $C^*(\mathcal{B}, \mathcal{L}, \theta)$ of a Boolean dynamical system $(\mathcal{B}, \mathcal{L}, \theta)$, that generalizes the $C^*$-algebra associated to a labelled graph introduced by Bates and Pask, and to determine its simplicity, its gauge invariant ideals, as well as compute its K-Theory. This is a joint work with Toke Meier Carlsen (Department of Science and Technology, University of the Faroe Islands) and Eduard Ortega (Department of Mathematical Sciences, NTNU Trondheim).Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Finite projections in multiplier algebras

We give a characterization of finiteness of projections in the multiplier algebra of a σ-unital C∗-algebra of realran k zero and stable rank one

Comparability, Separativity, and exchange rings

There are several long-standing open problems which ask whether regular rings, and C -algebras of real rank zero, satisfy certain module cancellation properties. Ara, Goodearl, O'Meara and Pardo recently observed that both types of rings are exchange rings, and showed that separative exchange rings have these good cancellation properties, thus answering the questions a rmatively in the separative case. In this ar- ticle, we prove that, for any positive integer s, exchange rings satisfying s-comparability are separative, thus answering the questions a rma- tively in the s-comparable case. We also introduce the weaker, more technical, notion of generalized s-comparability, and show that this condition still implies separativity for exchange rings. On restricting to directly nite regular rings, we recover results of Ara, O'Meara and Tyukavkin

Metric completions of ordered groups and K0+of exchange rings

We give a description of the closure of the natural a ne contin- uous function representation of K0(R) for any exchange ring R. This goal is achieved by extending the results of Goodearl and Handelman, about metric completions of dimension groups, to a more general class of pre-ordered groups, which includes K0 of exchange rings. As a consequence, the results about K+ 0 of regular rings, which the author gave in an earlier paper, can be extended to a wider class of rings, which includes C -algebras of real rank zero, among others. Also, the framework of pre-ordered groups developed here allows other potential applications

Dilations and full corners on fractional skew monoid rings

In this note we will show that the dilation result obtained for fractional skew monoid rings, in the case of a cancellative left Ore monoid S acting on a unital ring A by corner isomorphisms, holds in full generality. We apply this result to the context of semigroup C -crossed products

Embedding rank one simple groups into rank one simple Riesz groups

We give a method for embedding a large family of partially ordered simple groups of rank one into simple Riesz groups of rank one. In particular, we answer in the affirmative a question of Wehrung, by constructing a torsion-free, simple Riesz group G of rank one containing an interval D ¼= Gþ such that 2D ¼ Gþ. We sketch some potential applications of this result in the context of monoids of intervals and K-Theory of rings

On a density condition for K0+ of von Neumann regular rings

P.Ara and K.R.Goodearl, in [1], introduced and studied the concept of a regular ring R satisfying the following condition, which they called condition is dense in Aff(S(Ko(R)[R]))†, where Φ denotes the natural map from Ko(R) to Aff(S(Ko(R)[R])). They proved that every nonartinian, stably finite, strictly unperforated, simple regular ring satisfies condition (D). In this note we prove that a regular ring R satisfies condition (D) if and only if R has no nonzero artinian homomorphic image. We then obtain as a consequence that every nonartinian, simple regular ring satisfies condition (

The tight groupoid of the inverse semigroups of left cancellative small categories

We fix a path model for the space of filters of the inverse semigroup S_Λ associated to a left cancellative small category Λ. Then, we compute its tight groupoid, thus giving a representation of its C*-algebra as a (full) groupoid algebra. Using it, we characterize when these algebras are simple. Also, we determine amenability of the tight groupoid under mild, reasonable hypotheses.The second-named author was partially supported by PAI III grant FQM-298 of the Junta de Andalucía, and by the DGI-MINECO and European Regional Development Fund, jointly, through grant MTM2017-83487-P

Purely infinite simple skew group rings

In this note we prove that, if R is a purely infinite simple unital ring, G is a group, and α : G → Aut(R) is an outer action on R, then the skew group ring R∗α G is a purely infinite simple rin

Stable rank of Leavitt path algebras

We characterize the values of the stable rank for Leavitt path algebras by giving concrete criteria in terms of properties of the underlying graph