# Uniformly recurrent subgroups and the ideal structure of reduced crossed products

@article{Kawabe2017UniformlyRS, title={Uniformly recurrent subgroups and the ideal structure of reduced crossed products}, author={Takuya Kawabe}, journal={arXiv: Operator Algebras}, year={2017} }

We study the ideal structure of reduced crossed product of topological dynamical systems of a countable discrete group. More concretely, for a compact Hausdorff space $X$ with an action of a countable discrete group $\Gamma$, we consider the absence of a non-zero ideals in the reduced crossed product $C(X) \rtimes_r \Gamma$ which has a zero intersection with $C(X)$. We characterize this condition by a property for amenable subgroups of the stabilizer subgroups of $X$ in terms of the Chabauty…

## 17 Citations

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## References

SHOWING 1-10 OF 14 REFERENCES

Subgroup dynamics and $C^\ast$-simplicity of groups of homeomorphisms

- Mathematics
- 2016

We study the uniformly recurrent subgroups of groups acting by homeomorphisms on a topological space. We prove a general result relating uniformly recurrent subgroups to rigid stabilizers of the…

The ideal structure of reduced crossed products

- Mathematics
- 2010

Let (A, G) be a C*-dynamical system with G discrete. In this paper we in- vestigate the ideal structure of the reduced crossed product C*-algebra and in particular we determine sufficient—and in some…

Uniformly recurrent subgroups

- Mathematics
- 2014

We define the notion of uniformly recurrent subgroup, URS in short, which is a topological analog of the notion of invariant random subgroup (IRS), introduced in a work of M. Abert, Y. Glasner and B.…

Cartan Subalgebras in $C^*$-Algebras

- MathematicsIrish Mathematical Society Bulletin
- 2008

According to J. Feldman and C. Moore's well- known theorem on Cartan subalgebras, a variant of the group measure space construction gives an equivalence of categories between twisted countable…

SUBGROUP DYNAMICS AND C∗-SIMPLICITY OF GROUPS OF HOMEOMORPHISMS

- Mathematics
- 2016

We study the uniformly recurrent subgroups of groups acting by homeomorphisms on a topological space. We prove a general result relating uniformly recurrent subgroups to rigid stabilizers of the…

C*-simplicity and the unique trace property for discrete groups

- Mathematics
- 2014

A discrete group is said to be C*-simple if its reduced C*-algebra is simple, and is said to have the unique trace property if its reduced C*-algebra has a unique tracial state. A dynamical…

C*-simplicity and the amenable radical

- Mathematics
- 2015

A countable group is C*-simple if its reduced C*-algebra is simple. It is well known that C*-simplicity implies that the amenable radical of the group must be trivial. We show that the converse does…

A New Look at C∗-Simplicity and the Unique Trace Property of a Group

- Mathematics
- 2015

We characterize when the reduced C∗-algebra of a non-trivial group has unique tracial state, respectively, is simple, in terms of Dixmier-type properties of the group C∗-algebra. We also give a…

Characterizations of C*-simplicity

- Mathematics
- 2015

We characterize discrete groups that are C*-simple, meaning that their reduced C*-algebra is simple. First, we prove that a discrete group is C*-simple if and only if it has Powers' averaging…

LECTURE ON THE FURSTENBERG BOUNDARY AND C∗-SIMPLICITY

- Mathematics
- 2014

This is a handout for the lecture at the domestic “Annual Meeting of Operator Theory and Operator Algebras” at Toyo university, 24–26 December 2014. In this note, we first review the theory of the…