The general linear group as a complete invariant for C*-algebras

In 1955 Dye proved that two von Neumann factors not of type I_2n are isomorphic (via a linear or a conjugate linear *-isomorphism) if and only if their unitary groups are isomorphic as abstract groups. We consider an analogue for C*-algebras. We show that the topological general linear group is a classifying invariant for simple, unital AH-algebras of slow dimension growth and of real rank zero, and the abstract general linear group is a classifying invariant for unital Kirchberg algebras in the UCT class.Comment: 23 page

Strong pure infiniteness of crossed products

Consider an exact action of discrete group $G$ on a separable $C^*$-algebra $A$. It is shown that the reduced crossed product $A\rtimes_{\sigma, \lambda} G$ is strongly purely infinite - provided that the action of $G$ on any quotient $A/I$ by a $G$-invariant closed ideal $I\neq A$ is element-wise properly outer and that the action of $G$ on $A$ is $G$-separating (cf. Definition 4.1). This is the first non-trivial sufficient criterion for strong pure infiniteness of reduced crossed products of $C^*$-algebras $A$ that are not $G$-simple. In the case $A=\mathrm{C}_0(X)$ the notion of a $G$-separating action corresponds to the property that two compact sets $C_1$ and $C_2$, that are contained in open subsets $C_j\subseteq U_j \subseteq X$, can be mapped by elements of $g_j\in G$ onto disjoint sets $\sigma_{g_j}(C_j)\subseteq U_j$, but we do not require that $\sigma_{g_j}(U_j)\subseteq U_j$. A generalization of strong boundary actions on compact spaces to non-unital and non-commutative $C^*$-algebras $A$ (cf. Definition 6.1) is also introduced. It is stronger than the notion of $G$-separating actions by Proposition 6.6, because $G$-separation does not imply $G$-simplicity and there are examples of $G$-separating actions with reduced crossed products that are stably projection-less and non-simple.Comment: 30 pages, parts were taken out and included elsewher

Unbounded quasitraces, stable finiteness and pure infiniteness

We prove that if A is a \sigma-unital exact C*-algebra of real rank zero, then every state on K_0(A) is induced by a 2-quasitrace on A. This yields a generalisation of Rainone's work on pure infiniteness and stable finiteness of crossed products to the non-unital case. It also applies to k-graph algebras associated to row-finite k-graphs with no sources. We show that for any k-graph whose C*-algebra is unital and simple, either every twisted C*-algebra associated to that k-graph is stably finite, or every twisted C*-algebra associated to that k-graph is purely infinite. Finally we provide sufficient and necessary conditions for a unital simple k-graph algebra to be purely infinite in terms of the underlying k-graph.Comment: 38 page

Poland's Parliamentary Elections and a Looming Hungarian Scenario

Thanks to economic growth, Poland’s ruling PiS party has introduced social programs that have further bolstered its popularity. Unlike in recent European elections, the opposition is not running as a unified bloc in parliamentary elections on October 13, 2019. If PiS again wins a majority, it will take steps to cement its system of illiberal democracy. As long as he maintains good relations with Donald Trump, PiS’s leader Jaroslaw Kaczynski does not seem wary of reactions from Brussels and Berlin

The primitive ideal space of the C*-algebra of the affine semigroup of algebraic integers

We give a complete description of the primitive ideal space of the C*-algebra associated to the ring of integers R in a number field K as considered in a recent paper by Cuntz, Deninger and Laca

Purely infinite partial crossed products

Let (A,G,\alpha) be a partial dynamical system. We show that there is a bijective correspondence between G-invariant ideals of A and ideals in the partial crossed product A xr G provided the action is exact and residually topologically free. Assuming, in addition, a technical condition---automatic when A is abelian---we show that A xr G is purely infinite if and only if the positive nonzero elements in A are properly infinite in A xr G. As an application we verify pure infiniteness of various partial crossed products, including realisations of the Cuntz algebras O_n, O_A, O_N, and O_Z as partial crossed products.Comment: 30 page