相关论文: Topological boundaries of unitary representations

A discrete group $\Gamma$ is C*-simple if the C*-algebra $C_\lambda^*(\Gamma)$ generated by the range of the left regular representation $\lambda$ on $\ell^2(\Gamma)$ is simple. In this case, $\Gamma$ acts faithfully on the Furstenberg…

We investigate the tracial and ideal structures of $C^*$-algebras of quasi-regular representations of stabilizers of boundary actions. Our main tool is the notion of boundary maps, namely $\Gamma$-equivariant unital completely positive maps…

For a countable discrete group {\Gamma} and a minimal {\Gamma}-space X, we study the notion of ({\Gamma}, X)-boundary, which is a natural generalization of the notion of topological {\Gamma}-boundary in the sense of Furstenberg. We give…

We extend applications of Furstenberg boundary theory to the study of $C^*$-algebras associated to minimal actions $\Gamma\!\curvearrowright\! X$ of discrete groups $\Gamma$ on locally compact spaces $X$. We introduce boundary maps on…

Furstenberg has associated to every topological group $G$ a universal boundary $\partial(G)$. If we consider in addition a subgroup $H<G$, the relative notion of $(G,H)$-boundaries admits again a maximal object $\partial(G,H)$. In the case…

We develop a relative boundary theory for actions of discrete groups on compact spaces and use it to derive rigidity results for reduced crossed products. For a discrete group $\Gamma$ acting on a compact space $X$ and a subgroup $H$, we…

Consider a unitary representation $\pi$ of a discrete group $G$, which, when restricted to an almost normal subgroup $\Gamma\subseteq G$, is of type II. We analyze the associated unitary representation $\overline{\pi}^{\rm{p}}$ of $G$ on…

Let G be an amenable group, let X be a Banach space and let \pi : G --> B(X) be a bounded representation. We show that if the set {\pi(t) : t \in G} is gamma-bounded then \pi extends to a bounded homomorphism w : C*(G) --> B(X) on the group…

Let $G$ be a discrete countable group, and let $\Gamma$ be an almost normal subgroup. In this paper we investigate the classification of (projective) unitary representations $\pi$ of $G$ into the unitary group of the Hilbert space…

Poisson boundary is a measurable $\Gamma$-space canonically associated with a group $\Gamma$ and a probability measure $\mu$ on it. The collection of all measurable $\Gamma$-equivariant quotients, known as $\mu$-boundaries, of the Poisson…

Let $\Gamma$ be a Gromov hyperbolic group, endowed with an arbitrary left-invariant hyperbolic metric, quasi-isometric to a word metric. The action of $\Gamma$ on its boundary $\partial\Gamma$ endowed with the Patterson-Sullivan measure…

We clarify the relation between noncommutative Poisson boundaries and Furstenberg-Hamana boundaries of quantum groups. Specifically, given a compact quantum group $G$, we show that in many cases where the Poisson boundary of the dual…

For a locally compact quantum group $\mathbb{G}$, a (left) coideal is a (left) $\mathbb{G}$-invariant von Neumann subalgebra of $L^\infty(\mathbb{G})$. We introduce and analyze various generalizations of amenability and coamenability to…

We prove that a C$^*$-algebra $A$ has uniform property $\Gamma$ if the set of extremal tracial states, $\partial_e T(A)$, is a non-empty compact space of finite covering dimension and for each $\tau \in \partial_e T(A)$, the von Neumann…

We associate a boundary $\mathcal B_{\pi,u}$ to each covariant representation $(\pi,u,H)$ of $C^*$-dynamical system $(G,A,\alpha)$ and study the action of $G$ on $\mathcal B_{\pi,u}$ and its amenability properties. We relate rigidity…

We introduce and study the notions of boundary actions and of the Furstenberg boundary of a discrete quantum group. As for classical groups, properties of boundary actions turn out to encode significant properties of the operator algebras…

Let $\mathbf{G}$ be a connected reductive algebraic group over a $p$-adic local field $F$. In this paper we study the asymptotic behaviour of the trace characters $\theta _{\pi}$ evaluated at a regular element $\gamma $ of $\mathbf{G}(F)$…

We obtain an operator algebraic characterization of the noncommutative Furstenberg-Poisson boundary $\operatorname{L}(\Gamma) \subset \operatorname{L}(\Gamma \curvearrowright B)$ associated with an admissible probability measure $\mu \in…

This paper studies injective envelopes of groupoid dynamical systems and the corresponding boundaries. Analogue to the group case, we associate a bundle of compact Hausdorff spaces to any (discrete) groupoid (the Hamana boundary of the…

We consider symmetric random walks on discrete, Zariski-dense subgroups $\Gamma$ of a semisimple Lie group $G$ with Property (T). We prove that if $\Gamma$ has infinite covolume, then the associated hitting measure on the Furstenberg…