Related papers: $\mathrm{C}^*$-exactness and property A for group …

We introduce $C^*$-algebras associated with directed graphs, along with two generalizations of this concept, namely Exel-Pardo $C^*$-algebras associated with a self-similar action of a group on a directed graph, and the $C^*$-algebras…

Let $E = (E^0, E^1, r, s)$ be a topological graph with no sinks such that $E^0$ and $E^1$ are compact. We show that when $C^*(E)$ is finite, there is a natural isomorphism $C^*(E) \cong C(E^\infty) \rtimes \mathbb{Z}$, where $E^\infty$ is…

We show that the following properties of unital ${\rm C^*}$-algebra in a class of $\Omega$ are preserved by unital simple ${\rm C^*}$-algebra in the class of $\rm WTA\Omega$: $(1)$ uniform property $\Gamma$, $(2)$ a certain type of tracial…

We show by construction that when $G$ is an elementary amenable group and $A$ is a unital simple nuclear and tracially approximately divisible $C^*$-algebra, there exists an action $\omega$ of $G$ on $A$ with the tracial Rokhlin property in…

The goal of this paper is to study when uniform Roe algebras have certain $C^*$-algebraic properties in terms of the underlying space: in particular, we study properties like having stable rank one or real rank zero that are thought of as…

Given an action of a compact quantum group on a unital C*-algebra, one can amplify the action with an adjoint representation of the quantum group on a finite dimensional matrix algebra, and consider the resulting inclusion of fixed point…

Let $\Gamma$ be a simple undirected graph on a finite vertex set and let $A$ be its adjacency matrix. Then $\Gamma$ is {\it singular} if $A$ is singular. The problem of characterising singular graphs is easy to state but very difficult to…

A Cayley digraph $\Gamma$ over a finite group $G$ is said to be CI if for every Cayley digraph $\Gamma^\prime$ over $G$ isomorphic to $\Gamma$, there is an isomorphism from $\Gamma$ to $\Gamma^\prime$ which is at the same time an…

We investigate conditions on a graph $C^*$-algebra for the existence of a faithful semifinite trace. Using such a trace and the natural gauge action of the circle on the graph algebra, we construct a smooth $(1,\infty)$-summable semfinite…

We address the classification problem for graph $C^*$-algebras of finite graphs (finitely many edges and vertices), containing the class of Cuntz-Krieger algebras as a prominent special case. Contrasting earlier work, we do not assume that…

Every self-similar group acts on the space $X^\omega$ of infinite words over some alphabet $X$. We study the Schreier graphs $\Gamma_w$ for $w\in X^\omega$ of the action of self-similar groups generated by bounded automata on the space…

We prove that if $X$ is a compact, oriented, connected $4$-dimensional smooth manifold, possibly with boundary, satisfying $\chi(X)\neq 0$, then there exists an integer $C\geq 1$ such that any finite group $G$ acting smoothly and…

We investigate the ideal structures of the C^*-algebras arising from topological graphs. We give the complete description of ideals of such C^*-algebras which are invariant under the so-called gauge action, and give the condition on…

We show that the following conditions on a C*-algebra are equivalent: (i) it has the fixed point property for nonexpansive mappings, (ii) the spectrum of every self adjoint element is finite, (iii) it is finite dimensional. We prove that…

Let $\Gamma$ be a countable group and $(X, \Gamma)$ a compact topological dynamical system. We study the question of the existence of an intermediate $C^*$-subalgebra $\mathcal{A}$ $$C^{*}_{r}(\Gamma)<\mathcal{A}<C(X)\rtimes_r\Gamma,$$…

We present an axiomatic frame (in Prt I of this book) in which many results of the K-theory for C*-algebras are proved. Then we construct an example for this axiomatic theory (in Part II), which generalizes the classical theory for…

We characterize when the reduced C*-algebra of a group has unique tracial state, respectively, is simple, in terms of Dixmier-type properties of the group C*-algebra. We also give a simple proof of the recent result by Breuillard, Kalantar,…

We show that certain extensions of classifiable C*-algebra are strongly classified by the associated six-term exact sequence in K-theory together with the positive cone of K_{0}-groups of the ideal and quotient. We apply our result to give…

Given a $C^*$-dynamical system $(A,G,\alpha)$, with $G$ a discrete group, Schur $A$-multipliers and Herz--Schur $(A,G,\alpha)$-multipliers are used to implement approximation properties, namely exactness and the strong operator…

We introduce a family of $C^*$-correspondences $X_\alpha$ naturally associated to every ordinal graph $\Lambda$. When $\Lambda$ is a directed graph, $X_0$ is isomorphic to the usual $C^*$-correspondence associated to a graph. We show that…