arXiv 0709.36762007-09math.OAmath.GR

Haagerup逼近性质与相对顺从性

Haagerup's Approximation Property and Relative Amenability

Jon P. Bannon, Junsheng Fang

备注：: 16 pages

摘要：

具有忠实正规迹的有限von Neumann代数$\mathcal{M}$具有Haagerup逼近性质（相对于冯诺依曼子代数$\mathcal{N}$）如果存在正规完全正的网$(ϕ_α)_{α\in Λ}$（$\mathcal{N}$ -双模）从$\mathcal{M}$到$\mathcal{M}$的满足减法条件$% τ\circ ϕ_α\leq τ$的映射，扩张算子$% T_{ϕ_α}$是有界紧算子（在$<\mathcal{M%},e_{\mathcal{N}}>$中），并且逐点逼近迹范数中的恒等式，即，所有$% x\in \mathcal{M}$的$\lim_α||ϕ_α(x)-x||_{2}=0$。我们证明了减法条件可以删除，并提供了一个描述的Haagerup的近似性质的康纳斯的理论的对应。证明了如果$\mathcal{N}\subseteq \mathcal{M}$是有限von Neumann代数的顺从包含，且$% \mathcal{N}$具有Haagerup逼近性质，则$\mathcal{M}$也具有Haagerup逼近性质.这项工作回答了索林波帕的两个问题。

英文摘要：

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 that the subtraciality condition can be removed, and provide a description of Haagerup's approximation property in terms of Connes's theory of correspondences. We show that if $\mathcal{N}\subseteq \mathcal{M}$ is an amenable inclusion of finite von Neumann algebras and $% \mathcal{N}$ has Haagerup's approximation property, then $\mathcal{M}$ also has Haagerup's approximation property. This work answers two questions of Sorin Popa.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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