Separable boundaries for non-hyperbolic groups

Boundaries of Hyperbolic Metric Spaces

We investigate the relationship between the metric boundary and the Gromov boundary of a hyperbolic metric space. We show that the Gromov boundary is a quotient topological space of the metric boundary, and that therefore a word-hyperbolic group has an amenable action on the metric boundary of its Cayley graph. This result has significance for the study of Lip-norms on group C*-algebras.

$\mathcal Z$-stability and finite dimensional tracial boundaries

We show that a simple separable unital nuclear nonelementary $C^*$-algebra whose tracial state space has a compact extreme boundary with finite covering dimension admits uniformly tracially large order zero maps from matrix algebras into its central sequence algebra. As a consequence, strict comparison implies $\Z$-stability for these algebras.

Poisson boundaries over locally compact quantum groups

We present versions of several classical results on harmonic functions and Poisson boundaries in the setting of locally compact quantum groups. In particular, the Choquet--Deny theorem holds for compact quantum groups; also, the result of Kaimanovich--Vershik and Rosenblatt, which characterizes group amenability in terms of harmonic functions, admits a non-commutative analogue in the separable case. We also explore the relation between classical and quantum Poisson boundaries by investigating the spectrum of the quantum group. We apply this machinery to find a concrete realization of the Poiss…

相对双曲群上随机游走的比率极限边界

Ratio-limit boundaries for random walks on relatively hyperbolic groups

本文研究相对双曲群上随机游走的比率极限边界，扩展了Woess策略并揭示其局限，证明谱非退化情形下极小点闭包为唯一最小不变子空间，并推广了Toeplitz C*-代数共泛商的存在性，且构造了反例。

Unitarizable representations and fixed points of groups of biholomorphic transformations of operator balls

We show that the open unit ball of the space of operators from a finite dimensional Hilbert space into a separable Hilbert space (we call it "operator ball") has a restricted form of normal structure if we endow it with a hyperbolic metric (which is an analogue of the standard hyperbolic metric on the unit disc in the complex plane). We use this result to get a fixed point theorem for groups of biholomorphic automorphisms of the operator ball. The fixed point theorem is used to show that a bounded representation in a separable Hilbert space which has an invariant indefinite quadratic form with…