arXiv subjects
Publications and source records attributed to Yuhei Suzuki.
Recently Raum has given the first examples of locally compact non-discrete groups with the simple reduced group C*-algebra, answering a question of de la Harpe. Here we construct such groups whose proof relies only on results in the discrete case.
For an amenable minimal topologically free dynamical system $α$ of a group on a compact metrizable space $Z$ and for a compact metrizable space $Y$ satisfying a mild condition, we construct a minimal skew product extension of $α$ on $Z\times Y$. This generalizes a result of Glasner and Weiss. We also study the pure infiniteness of the crossed products of minimal dynamical systems arising from this result. In particular, we give a generalization of a result of Rørdam and Sierakowski.
We provide examples of ambient nuclear C*-algebras of non-nuclear C*-algebras with no proper intermediate C*-algebras. In particular this gives the first examples of minimal ambient nuclear C*-algebras of non-nuclear C*-algebras. For this purpose, we study generic Cantor systems of infinite free product groups.
We prove that for every exact discrete group $Γ$, there is an intermediate C*-algebra between the reduced group C*-algebra and the intersection of the group von Neumann algebra and the uniform Roe algebra which is realized as the intersection of a decreasing sequence of isomorphs of the Cuntz algebra $\mathcal{O}_2$. In particular, when $Γ$ has the AP (approximation property), the reduced group C*-algebra is realized in this way. We also study extensions of the reduced free group C*-algebras and show that any exact absorbing or unital absorbing extension of it by any stable separable nuclear C*-algebra is realized in this way.
It is proved that for every ICC group which is embeddable into a hyperbolic group, the reduced group C*-algebra is realized as the intersection of a decreasing sequence of isomorphs of the Cuntz algebra O_2. The proof is based on the study of amenable quotients of the boundary 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.
In this series of papers, we study correspondence between the following: (1) large scale structure of the metric space bigsqcup_m {Cay(G(m))} consisting of Cayley graphs of finite groups with k generators; (2) structure of groups which appear in the boundary of the set {G(m)}_m 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.
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 Popa's 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.