Related papers: Noncommutative topological boundaries and amenable…

We show that, for many choices of finite tuples of generators $X = (x_1, \dots , x_d)$ of a tracial von Neumann algebra $(M, \tau)$ satisfying certain decomposition properties (non-primeness, possessing a Cartan subalgebra, or property…

This is the second of two papers but has been written so as to have minimal dependence on the first paper (which is also on this archive). Let G be a group and let M be a CAT(0) proper metric space (e.g. a simply connected complete…

A topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann…

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…

Let G be one of the local gauge groups C(X,U(n)), C^\infty(X,U(n)), C(X,SU(n)) or C^\infty(X,SU(n)) where X is a compact Riemannian manifold. We observe that G has a nontrivial group topology, coarser than its natural topology, w.r.t. which…

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with vertex set $X$ and diameter $D$. Let $A$ denote the adjacency matrix of $\Gamma$. For a vertex $x\in X$ and for $0 \leq i \leq D$, let $E^*_i(x)$ denote the projection matrix…

We prove that, if a discrete group $G$ is not inner amenable, then the unit group of the ring of operators affiliated with the group von Neumann algebra of $G$ is non-amenable with respect to the topology generated by its rank metric. This…

Consider a lattice $\Gamma$ in a group $G = SL_2(\R), SO(1,n), SU(1,n)$, $SL_2(\Q_p)$. We discuss actions of $\Gamma$ by affine isometric transformations of Hilbert spaces. We show that for irreducible affine isometric action of $G$ its…

To any trace preserving action $\sigma: G \curvearrowright A$ of a countable discrete group on a finite von Neumann algebra $A$ and any orthogonal representation $\pi:G \to \mathcal O(\ell^2_{\mathbb{R}}(G))$, we associate the generalized…

Let $\Gamma \curvearrowright \Omega$ be a measure-preserving action and $\mathcal{L} \Gamma \hookrightarrow L^\infty(\Omega) \rtimes \Gamma$ the natural inclusion of the group von Neumann algebra into the crossed product. When $\mu(\Omega)…

Given a finite, directed, connected graph $\Gamma$ equipped with a weighting $\mu$ on its edges, we provide a construction of a von Neumann algebra equipped with a faithful, normal, positive linear functional…

We show that if $\Gamma\curvearrowright (X^\Gamma,\mu^\Gamma)$ is a Bernoulli action of an i.c.c. nonamenable group $\Gamma$ which is weakly amenable with Cowling-Haagerup constant $1$, and $\Lambda\curvearrowright(Y,\nu)$ is a free ergodic…

Given partitions $\alpha$, $\beta$, $\gamma$, the short exact sequences $0\to N_\alpha \to N_\beta \to N_\gamma \to 0$ of nilpotent linear operators of Jordan types $\alpha$, $\beta$, $\gamma$, respectively, define a constructible subset…

A topological group $G$ is extremely amenable if every continuous action of $G$ on a compact space has a fixed point. Using the concentration of measure techniques developed by Gromov and Milman, we prove that the group of automorphisms of…

The analytic von Neumann regular closure $R(\Gamma)$ of a complex group algebra $\C\Gamma$ was introduced by Linnell and Schick. This ring is the smallest $*$-regular subring in the algebra of affiliated operators $U(\Gamma)$ containing…

Motivated by Popa's seminal work \cite{Po04}, in this paper, we provide a fairly large class of examples of group actions $\Gamma \curvearrowright X$ satisfying the extended Neshveyev-St{\o}rmer rigidity phenomenon \cite{NS03}: whenever…

We introduce a new topological invariant, which is a nonnegative integer, of compact manifolds with boundaries associated with a kind of decomposition of them. Let M and N be m-dimensional compact connected manifolds with boundaries. The…

Let $(\Gamma,\mu)$ be a bipartite graph together with a weight on its vertices. Assume that $\mu$ is an eigenvector for the adjacency matrix of $\Gamma$. Let Aut$(\Gamma, \mu)$ be the automorphism group of the bipartite graph $\Gamma$ that…

We establish rigidity theorems for graph product von Neumann algebras $M_\Gamma=*_{v,\Gamma}M_v$ associated to finite simple graphs $\Gamma$ and families of tracial von Neumann algebras $(M_v)_{v\in\Gamma}$. We consider the following three…

Let $\Gamma$ be a countable discrete amenable group, and let $A=l^\infty(\Gamma) \rtimes \Gamma$ or $A = \mathrm{C}(M) \rtimes \Gamma$, where $(M, \Gamma)$ is the universal minimal set of $\Gamma$. It is shown that if $a, b \in A \otimes…