发表机构
IMPA（巴西纯数学与应用数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文从动力学角度研究一致Roe代数与拟局部代数的一致性，通过单参数自同构群刻画两者，并构造介于其间的粗等价不变$\mathrm{C}^*$-代数，证明该尺度非退化。
AI 中文摘要
一致Roe代数和拟局部代数都编码了度量空间的大尺度几何。这些代数是否一致的问题可追溯到Roe，并且最近才由Ozawa解决。我们对该问题提出一个动力学的观点。给定集合$X$，每个映射$h\colon X\to\mathbb{R}$诱导出$\mathcal{B}(\ell_2(X))$的自同构的单参数群$\sigma_h$，该群由对角酉算子$e^{ith}$的共轭作用给出，我们证明$\sigma_h$的连续性点算子恰好是$h$诱导的伪度量的一致Roe代数。若进一步假设$X$是均匀局部有限的度量空间，且$h$可取遍所有粗映射，则这给出两个代数的动力学刻画：$\mathrm{C}^*_{ql}(X)$恰好由对所有这样的流都连续的算子组成，而$\mathrm{C}^*_u(X)$恰好由那些解析算子的范数极限组成。介于连续性与解析性之间的正则性条件产生位于$\mathrm{C}^*_u(X)$与$\mathrm{C}^*_{ql}(X)$之间的$\mathrm{C}^*$-代数，且这些代数在双射粗等价下不变。我们证明这个尺度不是退化的：若$X$是展开图族的粗不交并，则由对每个对角流在条带上解析的算子生成的代数严格位于$\mathrm{C}^*_u(X)$与$\mathrm{C}^*_{ql}(X)$之间。
英文摘要
Both the uniform Roe algebras and the quasi-local algebras encode the large scale geometry of metric spaces. Whether these algebras coincide is a question which goes back to Roe and has only been recently solved by Ozawa. We propose a dynamical point of view on this problem. Given a set $X$, every map $h\colon X\to\mathbb{R}$ induces a one-parameter group $σ_h$ of automorphisms of $\mathcal{B}(\ell_2(X))$, given by conjugation by the diagonal unitaries $e^{ith}$, and we show that the operators which are continuity points of $σ_h$ are precisely the uniform Roe algebra of the pseudo-metric induced by $h$. If we moreover assume that $X$ is a uniformly locally finite metric space and $h$ is allowed to range over all coarse maps, this gives dynamical characterizations of both algebras: $\mathrm{C}^*_{ql}(X)$ consists exactly of the operators which are continuous for all such flows, while $\mathrm{C}^*_u(X)$ consists exactly of norm limits of those which are analytic. The regularity conditions lying between continuity and analyticity then give rise to $\mathrm{C}^*$-algebras between $\mathrm{C}^*_u(X)$ and $\mathrm{C}^*_{ql}(X)$ which are invariant under bijective coarse equivalence, and we show that this scale is not degenerate: if $X$ is a coarse disjoint union of expander graphs, the algebra generated by operators which are analytic on a strip for every diagonal flow sits strictly between $\mathrm{C}^*_u(X)$ and $\mathrm{C}^*_{ql}(X)$.
