Flows on uniform Roe algebras

Abstract

For a uniformly locally finite metric space $(X,d)$, we investigate \emph{coarse} flows on its uniform Roe algebra ${\mathrm{C}}_u^{\ast }(X)$, defined as one-parameter groups of automorphisms whose differentiable elements include all partial isometries arising from partial translations on $X$. We first show that any flow $\ddot{I}ƒ$ on ${\mathrm{C}}_u^{\ast }(X)$ corresponds to a (possibly unbounded) self-adjoint operator $h$ on ${\ell }_2(X)$ such that $\ddot{I}{ƒ}_t(a)=e^{ith}a{e}^{-ith}$ for all $t\in \mathbb{R}$, allowing us to focus on operators $h$ that generate flows on ${\mathrm{C}}_u^{\ast }(X)$. Assuming Yu's property A, we prove that a self-adjoint operator $h$ on ${\ell }_2(X)$ induces a coarse flow on ${\mathrm{C}}_u^{\ast }(X)$ if and only if $h$ can be expressed as $h=a+d$, where $a\in {\mathrm{C}}_u^{\ast }(X)$ and $d$ is a diagonal operator with entries forming a coarse function on $X$. We further study cocycle equivalence and cocycle perturbations of coarse flows, showing that, under property A, any coarse flow is a cocycle perturbation of a diagonal flow. Finally, for self-adjoint operators $h$ and $k$ that induce coarse flows on ${\mathrm{C}}_u^{\ast }(X)$, we characterize conditions under which the associated flows are either cocycle perturbations of each other or cocycle conjugate. In particular, if $h-k$ is bounded, then the flow induced by $h$ is a cocycle perturbation of the flow induced by $k$.