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Register a new userAlmost finiteness and the small boundary property
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Working within the framework of free actions of countable amenable groups on compact metrizable spaces, we show that the small boundary property is equivalent to a density version of almost finiteness, which we call almost finiteness in measure, and that under this hypothesis the properties of almost finiteness, comparison, and $m$-comparison for some nonnegative integer $m$ are all equivalent. The proof combines an Ornstein-Weiss tiling argument with the use of zero-dimensional extensions which are measure-isomorphic over singleton fibres. These kinds of extensions are also employed to show that if every free action of a given group on a zero-dimensional space is almost finite then so are all free actions of the group on spaces with finite covering dimension. Combined with recent results of Downarowicz-Zhang and Conley-Jackson-Marks-Seward-Tucker-Drob on dynamical tilings and of Castillejos-Evington-Tikuisis-White-Winter on the Toms-Winter conjecture, this implies that crossed product C$^*$-algebras arising from free minimal actions of groups with local subexponential growth on finite-dimensional spaces are classifiable in the sense of Elliotts program. We show furthermore that, for free actions of countably infinite amenable groups, the small boundary property implies that the crossed product has uniform property $Gamma$, which under minimality confirms the Toms-Winter conjecture for such crossed products by the aforementioned work of Castillejos-Evington-Tikuisis-White-Winter.
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We initiate a study of maximal subgroups and maximal von Neumann subalgebras which have the Haagerup property. We determine maximal Haagerup subgroups inside $mathbb{Z}^2 rtimes SL_2(mathbb{Z})$ and obtain several explicit instances where maximal Haagerup subgroups yield maximal Haagerup subalgebras. Our techniques are on one hand based on group-theoretic considerations, and on the other on certain results on intermediate von Neumann algebras, in particular these allowing us to deduce that all the intermediate algebras for certain inclusions arise from groups or from group actions. Some remarks and examples concerning maximal non-(T) subgroups and subalgebras are also presented, and we answer two questions of Ge regarding maximal von Neumann subalgebras.
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 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 that the ideal intersection property is equivalent to the absence of essentially confined amenable sections of isotropy groups. For groupoids with compact space of units we moreover show that is equivalent to the uniqueness of equivariant pseudo-expectations and in the minimal case to an appropriate generalisation of Powers averaging property. A key technical idea underlying our results is a new notion of groupoid action on C*-algebras that includes the essential groupoid C*-algebra itself. By considering a relative version of Powers averaging property, we obtain new examples of C*-irreducible inclusions in the sense of R{o}rdam. These arise from the inclusion of the C*-algebra generated by a suitable group representation into a simple groupoid C*-algebra. This is illustrated by the example of the C*-algebra generated by the quasi-regular representation of Thompsons group T with respect to Thompsons group F, which is contained C*-irreducibly in the Cuntz algebra $mathcal{O}_2$.
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We show that every free continuous action of a countably infinite elementary amenable group on a finite-dimensional compact metrizable space is almost finite. As a consequence, the crossed products of minimal such actions are $mathcal{Z}$-stable and classified by their Elliott invariant.
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 Rainones 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.
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