Related papers: Almost finiteness, comparison, and tracial $\mathc…

To a large class of graphs of groups we associate a C*-algebra universal for generators and relations. We show that this C*-algebra is stably isomorphic to the crossed product induced from the action of the fundamental group of the graph of…

Let $A$ be a separable, unital, simple, $\mathcal{Z}$-stable, nuclear $C^*$-algebra, and let $\alpha\colon G\to \mathrm{Aut}(A)$ be an action of a discrete, countable, amenable group. Suppose that the orbits of the action of $G$ on $T(A)$…

We presented a systematic treatment of a Hilbert criterion for stability theory for an action of a real reductive group $G$ on a real submanifold $X$ of a K\"ahler manifold $Z$. More precisely, we suppose the action of a compact connected…

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 investigate the representation theory of the crossed-product C*-algebra associated to a compact group G acting on a locally compact space X when the stability subgroups vary discontinuously. Our main result applies when G has a principal…

We associate a $C^*$-algebra to a partial action of the integers acting on the base space of a vector bundle, using the framework of Cuntz--Pimsner algebras. We investigate the structure of the fixed point algebra under the canonical gauge…

We prove that the C*-algebra of a minimal diffeomorphism satisfies Blackadar's Fundamental Comparability Property for positive elements. This leads to the classification, in terms of K-theory and traces, of the isomorphism classes of…

We prove that every acylindrically hyperbolic group admits a minimal and extremely proximal action on a compact metrizable space. If there are no nontrivial finite normal subgroups, then the action is topologically free. This answers…

We develop a general theory of local stability up to belonging to an ideal (e.g. having measure zero). From a model-theoretic perspective, we prove a stationarity principle for almost stable formulas in this sense, and build a topological…

We explore various limit constructions for C*-algebras, such as composition series and inverse limits, in relation to the notions of real rank, stable rank, and extremal richness. We also consider extensions and pullbacks. We identify some…

Let $A$ be a simple separable exact $C^*$-algebra that has traces. We show the following existed regularity properties are equivalent: \quad(1) $l^\infty(A)/J_A$ has real rank zero, where $J_A$ is the trace kernel ideal. \quad(2) $A$ is…

We examine the question of quasidiagonality for C*-algebras of discrete amenable groups from a variety of angles. We give a quantitative version of Rosenberg's theorem via paradoxical decompositions and a characterization of…

We give a rigorous account and prove continuity properties for the correspondence between almost flat bundles on a triangularizable compact connected space and the quasi-representations of its fundamental group. For a discrete countable…

Generalizing Block and Weinberger's characterization of amenability we introduce the notion of uniformly finite homology for a group action on a compact space and use it to give a homological characterization of topological amenability for…

Let $G$ be a countable group. We introduce several equivalence relations on the set ${\rm Sub}(G)$ of subgroups of $G$, defined by properties of the quasi-regular representations $\lambda_{G/H}$ associated to $H\in {\rm Sub}(G)$ and compare…

We analyse the structure of the quotient $\mathrm{A}_\sim(\Gamma,X,\mu)$ of the space of measure-preserving actions of a countable discrete group by the relation of weak equivalence. This space carries a natural operation of convex…

We prove that for any compact zero-dimensional metric space $X$ on which an infinite countable amenable group $G$ acts freely by homeomorphisms, there exists a dynamical quasitiling with good covering, continuity, F{\o}lner and dynamical…

Let $A$ be a unital simple C*-algebra with tracial rank zero and $X$ be a compact metric space. Suppose that $h_1, h_2: C(X)\to A$ are two unital monomorphisms. We show that $h_1$ and $h_2$ are approximately unitarily equivalent if and only…

We examine the ranks of operators in semi-finite C*-algebras as measured by their densely defined lower semicontinuous traces. We first prove that a unital simple C*-algebra whose extreme tracial boundary is nonempty and finite contains…

Commability is the finest equivalence relation between locally compact groups such that $G$ and $H$ are equivalent whenever there is a continuous proper homomorphism $G \to H$ with cocompact image. Answering a question of Cornulier, we show…