Yuhei Suzuki

Amenable minimal Cantor systems of free groups arising from diagonal actions

We study amenable minimal Cantor systems of free groups arising from the diagonal actions of the boundary actions and certain Cantor systems. It is shown that every virtually free group admits continuously many amenable minimal Cantor systems whose crossed products are mutually non-isomorphic Kirchberg algebras in the UCT class (with explicitly determined K-theory). The technique developed in our study also enables us to compute the K-theory of certain amenable minimal Cantor systems. We apply it to the diagonal actions of the boundary actions and the products of the odometer transformations, and determine their K-theory. Then we classify them in terms of the topological full groups, continuous orbit equivalence, strong orbit equivalence, and the crossed products.

Haagerup property for C⁎-algebras and rigidity of C⁎-algebras with property (T)

Abstract We study the Haagerup property for C ⁎ -algebras. We first give new examples of C ⁎ -algebras with the Haagerup property. A nuclear C ⁎ -algebra with a faithful tracial state always has the Haagerup property, and the permanence of the Haagerup property for C ⁎ -algebras is established. As a consequence, the class of all C ⁎ -algebras with the Haagerup property turns out to be quite large. We then apply Popaʼs results and show the C ⁎ -algebras with property (T) have a certain rigidity property. Unlike the case of von Neumann algebras, for the reduced group C ⁎ -algebras of groups with relative property (T), the rigidity property strongly fails in general. Nevertheless, for some groups without nontrivial property (T) subgroups, we show a rigidity property in some cases. As examples, we prove the reduced group C ⁎ -algebras of the (non-amenable) affine groups of the affine planes have a rigidity property.

Group approximation in Cayley topology and coarse geometry, III: Geometric property (T)

In this series of papers, we study the correspondence between the following: (1) the large scale structure of the metric space F m Cay.G .m/ / consisting of Cayley graphs of finite groups with k generators; (2) the structure of groups that appear in the boundary of the setfG .m/ g in the space of k‐marked groups. In this third part of the series, we show the correspondence among the metric properties “geometric property .T/”, “cohomological property .T/” and the group property “Kazhdan’s property .T/”. Geometric property .T/ of Willett‐Yu is stronger than being expander graphs. Cohomological property .T/ is stronger than geometric property .T/ for general coarse spaces. 20F65; 46M20

Almost Finiteness for General Étale Groupoids and Its Applications to Stable Rank of Crossed Products

We extend Matuis notion of almost finiteness to general etale groupoids and show that the reduced groupoid C*-algebras of minimal almost finite groupoids have stable rank one. The proof follows a new strategy, which can be regarded as a local version of the large subalgebra argument. The following three are the main consequences of our result. (i) For any group of (local) subexponential growth and for any its minimal action admitting a totally disconnected free factor, the crossed product has stable rank one. (ii) Any countable amenable group admits a minimal action on the Cantor set all whose minimal extensions form the crossed product of stable rank one. (iii) For any amenable group, the crossed product of the universal minimal action has stable rank one.

Group C*-algebras as decreasing intersection of nuclear C*-algebras

We prove that for every exact discrete group

Construction of minimal skew products of amenable minimal dynamical systems

\Gamma

Minimal ambient nuclear C⁎-algebras

, there is an intermediate

Simple equivariant C*-algebras whose full and reduced crossed products coincide

{\rm C}^\ast

Rigid sides of approximately finite dimensional simple operator algebras in non-separable category

-algebra between the reduced group

Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems.

{\rm C}^\ast