Flows on Uniform Roe Algebras

Abstract For a uniformly locally finite metric space (X, d), we investigate coarse flows on its uniform Roe algebra C^*ᵤ (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 on C^*ᵤ (X) corresponds to a (possibly unbounded) self-adjoint operator h on ₂ (X) such that ₜ (a) = e^ith a e^-ith for all t R, allowing us to focus on operators h that generate flows on C^*ᵤ (X). Assuming Yu’s property A, we prove that a self-adjoint operator h on ₂ (X) induces a coarse flow on C^*ᵤ (X) if and only if h can be expressed as h = a + d, where a C^*ᵤ (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 C^*ᵤ (X), we characterize conditions under which the associated flows are either cocycle perturbations of each other or cocycle conjugate to each other. In particular, if h - k is bounded, then the flow induced by h is a cocycle perturbation of the flow induced by k.