Amenability, Bilipschitz Maps, and the Von Neumann conjecture

On co-amenability for groups and von Neumann algebras

We first show that co-amenability does not pass to subgroups, answering a question asked by Eymard in 1972. We then address co-amenability for von Neumann algebras, describing notably how it relates to the former.

群冯诺依曼代数、内顺从性与连续环的单位群

Group von Neumann algebras, inner amenability, and unit groups of continuous rings

本文证明非内顺从离散群 $G$ 的群冯诺依曼代数相伴算子环的单位群在 rank 度量拓扑下非顺从，由此构造出非顺从单位群的非离散不可约连续环实例，并建立与 $\mathrm{II}_{1}$ 因子酉群 Eymard--Greenleaf 顺从性的联系。

A groupoid approach to Lück's amenability conjecture

We prove that amenability of a discrete group is equivalent to dimension flatness of certain ring inclusions naturally associated with measure preserving actions of the group. This provides a group-measure space theoretic solution to a conjecture of Lück stating that amenability of a group is characterized by dimension flatness of the inclusion of its complex group algebra into the associated von Neumann algebra.

Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups

We provide a general criterion to deduce maximal amenability of von Neumann subalgebras $LΛ\subset LΓ$ arising from amenable subgroups $Λ$ of discrete countable groups $Γ$. The criterion is expressed in terms of $Λ$-invariant measures on some compact $Γ$-space. The strategy of proof is different from S. Popa's approach to maximal amenability via central sequences [Po83], and relies on elementary computations in a crossed-product C*-algebra.

Haagerup's Approximation Property and Relative Amenability

A finite von Neumann algebra $\mathcal{M}$ with a faithful normal trace $% τ$ has Haagerup's approximation property (relative to a von Neumann subalgebra $\mathcal{N}$) if there exists a net $(ϕ_α)_{α\in Λ}$ of normal completely positive ($\mathcal{N}$-bimodular) maps from $\mathcal{M}$ to $\mathcal{M}$ that satisfy the subtracial condition $% τ\circ ϕ_α\leq τ$, the extension operators $% T_{ϕ_α}$ are bounded compact operators (in $<\mathcal{M%},e_{\mathcal{N}}>$), and pointwise approximate the identity in the trace-norm, i.e., $\lim_α||ϕ_α(x)-x||_{2}=0$ for all $% x\in \mathcal{M}$. We prove …

冯·诺依曼等价与群逼近性质

Von Neumann equivalence and group approximation properties

本文证明弱顺从性、弱Haagerup性质和逼近性质(AP)在冯·诺依曼等价下保持不变，并给出Haagerup性质不变性的新证明。