论文标题
测量的渐近扩张器和ROE代数的刚性
Measured asymptotic expanders and rigidity for Roe algebras
论文作者
论文摘要
我们关于ROE代数刚度的主要结果是：如果$ x $和$ y $是具有有界几何形状的公制空间，以至于它们的ROE代数为$*$ - 同构 - $ x $ y $ y $ y $ are coarsely均等，前提是$ x $或$ y $，只要不包含包含幽灵型扩展的sparce spectional assmptotic assmpt aspt aspt aspt aspt aspt aspt aspt aspt aspt aspt aspt aspt aspt aspt aspt aspt aspt aspt aspt aspt aspt aspt aspt aspt aspt aspt aspt aSpt ospts。请注意，这种几何条件概括了用于ROE代数刚度的现有技术假设。 因此，我们证明了所有有界几何空间的刚度，这些空间嵌入了一些$ l^p $ - 空间中的$ p \ in [1，\ infty）$。此外，我们还验证了由Arzhantseva-Tessera和Delabie-Khukhro构建的盒子空间的刚度，即使它们做\ emph {not}将其粗糙地嵌入任何$ l^p $ -Space中。 朝着刚性证明的关键步骤是证明稀疏空间上的块级（ghost）投影$ x $属于Roe代数$ C^*（x）$，并且仅当$ x $由（ghostly）测量的渐近扩张器组成。作为副产品，我们还推断出幽灵测得的渐近扩张器是对粗糙的鲍姆 - 康纳斯猜想的反例的新来源。
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 also verify the rigidity for the box spaces constructed by Arzhantseva-Tessera and Delabie-Khukhro even though they do \emph{not} coarsely embed into any $L^p$-space. The key step towards our proof for the rigidity is to show that a block-rank-one (ghost) projection on a sparse space $X$ belongs to the Roe algebra $C^*(X)$ if and only if $X$ consists of (ghostly) measured asymptotic expanders. As a by-product, we also deduce that ghostly measured asymptotic expanders are new sources of counterexamples to the coarse Baum-Connes conjecture.
