拟局部代数与渐近展开式

关于扩张图的准局部代数的注记

A note on the quasi-local algebra of expander graphs

本文研究了扩张图的准局部代数，证明其不包含与$\ell_\infty$同构的Cartan子代数，并指出其与均匀Roe代数的区别。

On the structure of asymptotic expanders

In this paper, we use geometric tools to study the structure of asymptotic expanders and show that a sequence of asymptotic expanders always admits a "uniform exhaustion by expanders". It follows that asymptotic expanders cannot be coarsely embedded into any $L^p$-space, and that asymptotic expanders can be characterised in terms of their uniform Roe algebra. Moreover, we provide uncountably many new counterexamples to the coarse Baum--Connes conjecture. These appear to be the first counterexamples that are not directly constructed by means of spectral gaps. Finally, we show that vertex-transi…

Measured asymptotic expanders and rigidity for Roe algebras

Our main result about rigidity of Roe algebras is the following: if $X$ and $Y$ are metric spaces with bounded geometry such that their Roe algebras are $*$-isomorphic, then $X$ and $Y$ are coarsely equivalent provided that either $X$ or $Y$ contains no sparse subspaces consisting of ghostly measured asymptotic expanders. Note that this geometric condition generalises the existing technical assumptions used for rigidity of Roe algebras. Consequently, we show that the rigidity holds for all bounded geometry spaces which coarsely embed into some $L^p$-space for $p\in [1,\infty)$. Moreover, we al…

Asymptotic dimension and coarse embeddings in the quantum setting

We generalize the notions of asymptotic dimension and coarse embeddings from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver. We show that quantum asymptotic dimension behaves well with respect to metric quotients and direct sums, and is preserved under quantum coarse embeddings. Moreover, we prove that a quantum metric space that equi-coarsely contains a sequence of reflexive quantum expanders must have infinite asymptotic dimension. This is done by proving a quantum version of a vertex-isoperimetric inequality for expanders, based upon a previously known edge-isop…

几乎拟等距与更多的非正合群

Almost quasi-isometries and more non-exact groups

我们研究持久性的结果几乎拟等距，所产生的地图从Gromov建设的随机群，包含扩展（因此，不是C*-精确）。本文证明了在几乎拟等距下，一列围长大且顶点度可控的有限图的象不具有余国梁的性质A。我们用这个结果来扩大应用Gromov的技术序列的图形不一定是扩展。这样，我们得到了更多的非C *-正合群的例子。

量子展开子与算子空间几何

Quantum Expanders and Geometry of Operator Spaces

我们发现，有很好的分离家庭的量子膨胀剂渐近的最大基数允许一个已知的上限。这适用于某些算子空间的“增长”：它意味着对$M_N$ -空间的多重性的增长的渐近尖锐估计，所述$M_N$ -空间需要将$n$ -维算子希尔伯特空间$OH_n$的$M_N$ -版本表示为$M_N$的副本的直和。我们发现，当$C$接近1时，对于某些常数$β>0$，这种多重性增长为$\exp{βn N^2}$。其主要思想是将量子展开子与欧几里得单位球的矩阵类似物上的“光滑”点联系起来。这将$n$维Hilbert空间（对应于N=1）上的一个经典几何结果推广到了算子空间.在附录中，我们给出了随机酉矩阵上的一个不等式（与Hastings之前的工作相关）的快速证明，这对本文至关重要。