arXiv subjects

Dan Ursu

Publications and source records attributed to Dan Ursu.

6 recordsLinked to original sources

arXiv2024

Intermediate subalgebras for reduced crossed products of discrete groups

Let $\alpha : \Gamma \curvearrowright A$ be an action of a discrete group $\Gamma$ on a unital C*-algebra $A$ by *-automorphisms and let $A \rtimes_{\alpha,\lambda} \Gamma$ denote the corresponding reduced crossed product C*-algebra. Assuming that $\Gamma$ satisfies the approximation property, we establish a sufficient and (almost always) necessary condition on the action $\alpha$ for the existence of a Galois correspondence between intermediate C*-algebras for the inclusion $A \subseteq A \rtimes_{\alpha,\lambda} \Gamma$ and partial subactions of $\alpha$. This condition, which we refer to as pointwise residual proper outerness, is a natural noncommutative generalization of freeness.

math.OA↗

arXiv2023

Simplicity of crossed products by FC-hypercentral groups

In this paper, we give a complete, two-way characterization, of when a noncommutative crossed product $A \rtimes_\lambda G$ is simple, in the case of $G$ being an FC-hypercentral group. This is a large class of amenable groups that contains all virtually nilpotent groups, and in the finitely-generated setting, coincides with the set of groups which have polynomial growth. We further completely characterize the ideal intersection property under the assumption that the group is FC, meaning that every element has a finite conjugacy class. Finally, for minimal actions of arbitrary discrete groups on unital C*-algebras, we are able to characterize when the crossed product $A \rtimes_\lambda G$ is prime.

math.OA↗

arXiv2021

Characterizing traces on crossed products of noncommutative C*-algebras

We give complete descriptions of the tracial states on both the universal and reduced crossed products of a C*-dynamical system consisting of a unital C*-algebra and a discrete group. In particular, we also answer the question of when the tracial states are in canonical bijection with the invariant tracial states on the original C*-algebra. This generalizes the unique trace property for discrete groups. The analysis simplifies greatly in various cases, for example when the conjugacy classes of the original group are all finite, and in other cases gives previously known results, for example when the original C*-algebra is commutative. We also obtain results and examples in the case of abelian groups that contradict existing results in the literature of Bédos and Thomsen. Specifically, we give a finite-dimensional counterexample, and provide a correction to the result of Thomsen.

math.OA↗

arXiv2021

The ideal intersection property for essential groupoid C*-algebras

We characterise, in several complementary ways, \'etale groupoids with locally compact Hausdorff space of units whose essential groupoid C*-algebra has the ideal intersection property, assuming that the groupoid is topologically transitive and either Hausdorff or $\sigma$-compact. This leads directly to a characterisation of the simplicity of this C*-algebra which, for Hausdorff groupoids, agrees with the reduced groupoid C*-algebra. Specifically, we prove for topologically transitive groupoids that the ideal intersection property is equivalent to the absence of essentially confined amenable sections of isotropy groups. For topologically transitive groupoids with compact space of units we moreover show that this is equivalent to the uniqueness of equivariant pseudo-expectations. A key technical idea underlying our results is a new notion of groupoid action on C*-algebras including the essential groupoid C*-algebra itself. For minimal groupoids, we further obtain a relative version of Powers averaging property. Examples arise from suitable group representations into simple groupoid \Cstar-algebras. This is illustrated by the example of the quasi-regular representation of Thompson's group $\mathrm{T}$ with respect to Thompson's group $\mathrm{F}$, which satisfies the relative Powers averaging property in the Cuntz algebra $\mathcal{O}_2$.

math.OA↗

arXiv2021

A generalized Powers averaging property for commutative crossed products

We prove a generalized version of Powers' averaging property that characterizes simplicity of reduced crossed products $C(X) \rtimes_\lambda G$, where $G$ is a countable discrete group, and $X$ is a compact Hausdorff space which $G$ acts on minimally by homeomorphisms. As a consequence, we generalize results of Hartman and Kalantar on unique stationarity to the state space of $C(X) \rtimes_\lambda G$ and to Kawabe's generalized space of amenable subgroups $\operatorname{Sub}_a(X,G)$. This further lets us generalize a result of the first named author and Kalantar on simplicity of intermediate C*-algebras. We prove that if $C(Y) \subseteq C(X)$ is an inclusion of unital commutative $G$-C*-algebras with $X$ minimal and $C(Y) \rtimes_\lambda G$ simple, then any intermediate C*-algebra $A$ satisfying $C(Y) \rtimes_\lambda G \subseteq A \subseteq C(X) \rtimes_\lambda G$ is simple.

math.OA↗

arXiv2019

Relative C*-simplicity and characterizations for normal subgroups

The notion of a plump subgroup was recently introduced by Amrutam. This is a relativized version of Powers' averaging property, and it is known that Powers' averaging property is equivalent to C*-simplicity. With this in mind, we introduce a relativized notion of C*-simplicity, and show that for normal subgroups it is equivalent to plumpness, along with several other characterizations.

math.OA↗