Hiroki Matui

SOME REMARKS ON TOPOLOGICAL FULL GROUPS OF CANTOR MINIMAL SYSTEMS

Giordano, Putnam and Skau showed that topological full groups of Cantor minimal systems are complete invariants for flip conjugacy. We will completely determine the structure of normal subgroups of the topological full group. Moreover, a necessary and sufficient condition for the topological full group to be finitely generated will be given.

Decomposition rank of UHF-absorbing

Let A be a unital separable simple C*-algebra with a unique tracial state. We prove that if A is nuclear and quasidiagonal, then A tensored with the universal UHF-algebra has decomposition rank at most one. Then it is proved that A is nuclear, quasidiagonal and has strict comparison if and only if A has finite decomposition rank. For such A, we also give a direct proof that A tensored with a UHF-algebra has tracial rank zero. Applying this characterization, we obtain a counter-example to the Powers-Sakai conjecture.

\mathrm{C}^{*}

For almost finite groupoids, we study how their homology groups reflect dynamical properties of their topological full groups. It is shown that two clopen subsets of the unit space has the same class in H_0 if and only if there exists an element in the topological full group which maps one to the other. It is also shown that a natural homomorphism, called the index map, from the topological full group to H_1 is surjective and any element of the kernel can be written as a product of four elements of finite order. In particular, the index map induces a homomorphism from H_1 to K_1 of the groupoid C^*-algebra. Explicit computations of homology groups of AF groupoids and etale groupoids arising from subshifts of finite type are also given.

-algebras

We show that every minimal action of any finitely generated abelian group on the Cantor set is (topologically) orbit equivalent to an AF relation. As a consequence, this extends the classification up to orbit equivalence of minimal dynamical systems on the Cantor set to include AF relations and ℤd-actions.

Homology and topological full groups of étale groupoids on totally disconnected spaces

We consider a crossed product of a unital simple separable nuclear stably finite

Orbit equivalence for Cantor minimal ℤd-systems

\cal Z

Z-STABILITY OF CROSSED PRODUCTS BY STRONGLY OUTER ACTIONS II

-stable

Continuous orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras

C^*

{\mathcal {Z}} -Stability of Crossed Products by Strongly Outer Actions

-algebra

Minimal Dynamical Systems on the Product of the Cantor Set and the Circle

A