论文标题
$ C^*$ - 代数的Schur乘数，群体不变的压实和适用于舒适性和渗透的应用
Schur Multipliers of $C^*$-algebras, group-invariant compactification and applications to amenability and percolation
论文作者
论文摘要
令$γ$为一个可计数的离散组。给定任何序列$（f_n）_ {n \ geq 1} $ of $ \ ell^p $ - 正态化函数（$ p \ in [1,2）$），考虑相关的正定矩阵系数$ \ langle f_n，ρ（\ cdot）f_n \ rangle $ primander代表$ρ。我们在还原的组$ C^*$ - 代数或$γ$的均匀的ROE代数上构建相应{\ it schur乘数}的正交分解。我们通过轨道的限制点$（\ widetilde f_n）_ {n \ geq 1} $明确识别该分解点，这是瓦拉达汉（Varadhan）和[14]中的第一作者的群 - invariant紧凑型。我们应用此结果并使用正定性，提供两个（完全不同的）$γ$的特征 - 一种是通过变异方法，另一种是在Benjamini，Lyons，Peres和Schramm [1]上使用群 - 不变的渗透性[1]。这些结果从新的角度到我们的最佳知识强调，Schur乘数捕获了基础组$γ$的几何特性的方式。
Let $Γ$ be a countable discrete group. Given any sequence $(f_n)_{n\geq 1}$ of $\ell^p$-normalized functions ($p\in [1,2)$), consider the associated positive definite matrix coefficients $\langle f_n, ρ(\cdot) f_n\rangle$ of the right regular representation $ρ$. We construct an orthogonal decomposition of the corresponding {\it Schur multipliers} on the reduced group $C^*$-algebra or the uniform Roe algebra of $Γ$. We identify this decomposition explicitly via the limit points of the orbits $(\widetilde f_n)_{n\geq 1}$ in the group-invariant compactification of the quotient space constructed by Varadhan and the first author in [14]. We apply this result and use positive-definiteness to provide two (quite different) characterizations of amenability of $Γ$ -- one via a variational approach and the other using group-invariant percolation on Cayley graphs constructed by Benjamini, Lyons, Peres and Schramm [1]. These results underline, from a new point of view to the best of our knowledge, the manner in which Schur multipliers capture geometric properties of the underlying group $Γ$.
