Related papers: Strict comparison for $C^*$-algebras arising from …
We prove that for every $n\geq 2$, the reduced group $C^*$-algebras of the countable free groups $C^*_r(\mathbb{F}_n)$ have strict comparison. Our method works in a general setting: for $G$ in a large family of non-amenable groups,…
The notion of almost elementariness for a locally compact Hausdorff \'{e}tale groupoid $\mathcal{G}$ with a compact unit space was introduced by the authors as a sufficient condition ensuring the reduced groupoid $C^*$-algebra…
In this paper, we study the ideal structure of reduced $C^*$-algebras $C^*_r(G)$ associated to \'etale groupoids $G$. In particular, we characterize when there is a one-to-one correspondence between the closed, two-sided ideals in…
We show that if $G$ is a second countable locally compact Hausdorff \'etale groupoid carrying a suitable cocycle $c:G\to\mathbb{Z}$, then the reduced $C^*$-algebra of $G$ can be realised naturally as the Cuntz-Pimsner algebra of a…
Suppose that A is a C*-algebra for which A is isomorphic to A tensor Z, where Z is the Jiang-Su algebra: a unital, simple, stably finite, separable, nuclear, infinite dimensional C*-algebra with the same Elliott invariant as the complex…
It is shown that every Jiang-Su stable approximately subhomogeneous C*-algebra has finite decomposition rank. Previously, it was not even known that such algebras have finite nuclear dimension. A key step in the proof is that subhomogeneous…
We study the finite versus infinite nature of C*-algebras arising from etale groupoids. For an ample groupoid G, we relate infiniteness of the reduced C*-algebra of G to notions of paradoxicality of a K-theoretic flavor. We construct a…
We prove that separable C*-algebras which are completely close in a natural uniform sense have isomorphic Cuntz semigroups, continuing a line of research developed by Kadison - Kastler, Christensen, and Khoshkam. This result has several…
Let $G$ be a Hausdorff, \'etale groupoid that is minimal and topologically principal. We show that $C^*_r(G)$ is purely infinite simple if and only if all the nonzero positive elements of $C_0(G^0)$ are infinite in $C_r^*(G)$. If $G$ is a…
We prove that faithful traces on separable and nuclear C*-algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear…
We give an example of an exact, stably finite, simple. separable C*-algebra D which is not isomorphic to its opposite algebra. Moreover, D has the following additional properties. It is stably finite, approximately divisible, has real rank…
In this paper, we exhibit two unital, separable, nuclear ${\rm C}^*$-algebras of stable rank one and real rank zero with the same ordered scaled total K-theory, but they are not isomorphic with each other, which forms a counterexample to…
We examine the question of quasidiagonality for C*-algebras of discrete amenable groups from a variety of angles. We give a quantitative version of Rosenberg's theorem via paradoxical decompositions and a characterization of…
We study C*-algebras associated with subsemigroups of groups. For a large class of such semigroups including positive cones in quasi-lattice ordered groups and left Ore semigroups, we describe the corresponding semigroup C*-algebras as…
We demonstrate that pure C*-algebras form a robust class by proving that pureness follows from very weak comparison and divisibility properties. Using this, we show that every simple, non-elementary C*-algebra with a unique quasitrace and…
A simple Steinberg algebra associated to an ample Hausdorff groupoid $G$ is algebraically purely infinite if and only if the characteristic functions of compact open subsets of the unit space are infinite idempotents. If a simple Steinberg…
We exhibit a countably infinite family of simple, separable, nuclear, and mutually non-isomorphic C*-algebras which agree on K-theory and traces. The algebras do not absorb the Jiang-Su algebra Z tensorially, answering a question of N. C.…
We prove that any reduced twisted group C*-algebra of a selfless group with the rapid decay property is selfless. As an application, we show that twisted group C*-algebras of acylindrically hyperbolic groups (possibly with nontrivial finite…
We study comparison properties in the category Cu aiming to lift results to the C*-algebraic setting. We introduce a new comparison property and relate it to both the CFP and $\omega$-comparison. We show differences of all properties by…
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…