On Ozawa kernels

Exactness of free and amenable groups by the construction of Ozawa kernels

Using properties of their Cayley graphs, specific examples of Ozawa kernels are constructed for both free and amenable groups, thus showing that these groups satisfy Property O. It is deduced both that these groups are exact and satisfy Yu's Property A.

非单位约化交叉积的简单性与迹权重

Simplicity and tracial weights on non-unital reduced crossed products

本文在温和假设下将单位约化交叉积的定理推广至非单位情形，证明了C*-代数简单性及迹权重唯一性在离散C*-单群上的稳定性，并应用于局部紧群扩张与扭曲交叉积。

Boundaries of reduced C*-algebras of discrete groups

For a discrete group G, we consider the minimal C*-subalgebra of $\ell^\infty(G)$ that arises as the image of a unital positive G-equivariant projection. This algebra always exists and is unique up to isomorphism. It is trivial if and only if G is amenable. We prove that, more generally, it can be identified with the algebra $C(\partial_F G)$ of continuous functions on Furstenberg's universal G-boundary $\partial_F G$. This operator-algebraic construction of the Furstenberg boundary has a number of interesting consequences. We prove that G is exact precisely when the G-action on $\partial_F G$…

Unitary representations of the Roe algebra of a discrete group and symmetries

Let $Γ$ be a discrete countable group. Consider the crossed product C$^\ast$-algebra $\mathfrak{R}(Γ) = C^{\ast}(Γ\rtimes l^{\infty}(Γ))$. Let $G$ be a larger discrete group, containing $Γ$ as an almost normal subgroup. Consequently $G$ acts by partial isomorphisms on $G$ and hence on $\mathfrak {R}(Γ)$. Let $\mathfrak{R}_G(Γ)$ be the crossed product $C^{\ast}$ - algebra $C^{\ast}(G \times (\mathfrak{R}(Γ))$. The C$^\ast$-algebra $\mathfrak{R}_G(Γ)$ has a natural representation into $\mathcal B(\ell ^2(Γ))$ and hence also admits a representation $Π_{\mathcal{Q}}$ into the Calkin algebra $\math…

Examples of group actions which are virtually W*-superrigid

We show that if G is a discrete group which does not have the Haagerup property but does have an unbounded cocycle into a C_0 representation and if G acts on a finite von Neumann algebra B such that the inclusion B \subset (B \rtimes G) has the Haagerup property from below then any group-measure space Cartan subalgebra must have a corner which embeds into B inside B \rtimes G. Taking the action to be trivial we produce examples of II_1 factors N such that N \otimes M is not a group-measure space construction whenever M is a finite factor with the Haagerup property. Taking the action on a proba…

Hyperlinear and sofic groups: a brief guide

Relatively recently, two new classes of (discrete, countable) groups have been isolated: hyperlinear groups and sofic groups. They come from different corners of mathematics (operator algebras and symbolic dynamics, respectively), and were introduced independently from each other, but are closely related nevertheless. Hyperlinear groups have their origin in Connes' Embedding Conjecture about von Neumann factors of type $II_1$, while sofic groups, introduced by Gromov, are motivated by Gottschalk Surjunctivity Conjecture (can a shift $A^G$ contain a proper isomorphic copy of itself, where $A$ i…