Dynamical ${\mathrm{C}}^{\ast }$-algebras and coarse geometry

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 $h\text{\hspace{0.17em}⁣}:X\to \mathbb{R}$ induces a one-parameter group ${\sigma }_h$ of automorphisms of $\mathcal{B}({\mathrm{\ell }}_2(X))$, given by conjugation by the diagonal unitaries $e^{ith}$, and we show that the operators which are continuity points of ${\sigma }_h$ 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: ${\mathrm{C}}_{ql}^{\ast }(X)$ consists exactly of the operators which are continuous for all such flows, while ${\mathrm{C}}_u^{\ast }(X)$ consists exactly of norm limits of those which are analytic. The regularity conditions lying between continuity and analyticity then give rise to ${\mathrm{C}}^{\ast }$-algebras between ${\mathrm{C}}_u^{\ast }(X)$ and ${\mathrm{C}}_{ql}^{\ast }(X)$ 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 ${\mathrm{C}}_u^{\ast }(X)$ and ${\mathrm{C}}_{ql}^{\ast }(X)$.