arXiv subjects
Publications and source records attributed to Adam Sierakowski.
We study the structure and compute the stable rank of C*-algebras of finite higher-rank graphs. We completely determine the stable rank of the C*-algebra when the k-graph either contains no cycle with an entrance, or is cofinal. We also determine exactly which finite, locally convex k-graphs yield unital stably finite C*-algebras. We give several examples to illustrate our results.
We establish exact sequences in $KK$-theory for graded relative Cuntz-Pimsner algebras associated to nondegenerate $C^*$-correspondences. We use this to calculate the graded $K$-theory and $K$-homology of relative Cuntz-Krieger algebras of directed graphs for gradings induced by $\{0,1\}$-valued labellings of their edge sets.
We introduce an algebraic version of the Katsura $C^*$-algebra of a pair $A,B$ of integer matrices and an algebraic version of the Exel-Pardo $C^*$-algebra of a self-similar action on a graph. We prove a Graded Uniqueness Theorem for such algebras and construct a homomorphism of the latter into a Steinberg algebra that, under mild conditions, is an isomorphism. Working with Steinberg algebras over non-Hausdorff groupoids we prove that in the unital case, our algebraic version of Katsura $C^*$-algebras are all isomorphic to Steinberg algebras.
We consider the general linear group as an invariant of von Neumann factors. We prove that up to complement, a set consisting of all idempotents generating the same right ideal admits a characterisation in terms of properties of the general linear group of a von Neumann factor. We prove that for two Neumann factors, any bijection of their general linear groups induces a bijection of their idempotents with the following additional property: If two idempotents or their two complements generate the same right ideal, then so does their image. This generalises work on regular rings, such include von Neumann factors of type $I_{n}$, $n < \infty$.
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.
From a suitable groupoid G, we show how to construct an amenable principal groupoid whose C*-algebra is a Kirchberg algebra which is KK-equivalent to C*(G). Using this construction, we show by example that many UCT Kirchberg algebras can be realised as the C*-algebras of amenable principal groupoids.
We study dimension theory for the $C^*$-algebras of row-finite $k$-graphs with no sources. We establish that strong aperiodicity - the higher-rank analogue of condition (K) - for a $k$-graph is necessary and sufficient for the associated $C^*$-algebra to have topological dimension zero. We prove that a purely infinite $2$-graph algebra has real-rank zero if and only if it has topological dimension zero and satisfies a homological condition that can be characterised in terms of the adjacency matrices of the $2$-graph. We also show that a $k$-graph $C^*$-algebra with topological dimension zero is purely infinite if and only if all the vertex projections are properly infinite. We show by example that there are strongly purely infinite $2$-graphs algebras, both with and without topological dimension zero, that fail to have real-rank zero.
We introduce filling families with matrix diagonalization as a refinement of the work by R{\o}rdam and the first named author. As an application we improve a result on local pure infiniteness and show that the minimal tensor product of a strongly purely infinite $C^*$-algebra and a exact $C^*$-algebra is again strongly purely infinite. Our results also yield a sufficient criterion for the strong pure infiniteness of crossed products $A\rtimes_\varphi \mathbb{N}$ by an endomorphism $\varphi$ of $A$ (cf. Theorem 7.6). Our work confirms that the special class of nuclear Cuntz-Pimsner algebras constructed by Harnisch and the first named author consist of strongly purely infinite $C^*$-algebras, and thus absorb $\mathcal{O}_\infty$ tensorially.
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 Hausdorff, ample groupoid, then we show that $C^*_r(G)$ is purely infinite simple if and only if every nonzero projection in $C_0(G^0)$ is infinite in $C^*_r(G)$. We then show how this result applies to $k$-graph $C^*$-algebras. Finally, we investigate strongly purely infinite groupoid $C^*$-algebras.
Given a system of coverings of k-graphs, we show that the cohomology of the resulting (k+1)-graph is isomorphic to that of any one of the k-graphs in the system. We then consider Bratteli diagrams of 2-graphs whose twisted C*-algebras are matrix algebras over noncommutative tori. For such systems we calculate the ordered K-theory and the gauge-invariant semifinite traces of the resulting 3-graph C*-algebras. We deduce that every simple C*-algebra of this form is Morita equivalent to the C*-algebra of a rank-2 Bratteli diagram in the sense of Pask-Raeburn-R{\o}rdam-Sims.
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.
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.
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.
We study conditions that will ensure that a crossed product of a C*-algebra by a discrete exact group is purely infinite (simple or non-simple). We are particularly interested in the case of a discrete non-amenable exact group acting on a commutative C*-algebra, where our sufficient conditions can be phrased in terms of paradoxicality of subsets of the spectrum of the abelian C*-algebra. As an application of our results we show that every discrete countable non-amenable exact group admits a free amenable minimal action on the Cantor set such that the corresponding crossed product C*-algebra is a Kirchberg algebra in the UCT class.
Let (A,G) be a C*-dynamical system with G discrete. In this paper we investigate the ideal structure of the reduced crossed product C*-algebra and in particular we determine sufficient - and in some cases also necessary - conditions for A to separate the ideals in Ax_rG. When A separates the ideals in Ax_rG, then there is a one-to-one correspondence between the ideals in Ax_rG and the invariant ideals in A. We extend the concept of topological freeness and present a generalization of the Rokhlin property. Exactness properties of (A,G) turns out to be crucial in these investigations.