FLOWS ON UNIFORM ROE ALGEBRAS

<jats:p> For a uniformly locally finite metric space <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748025101473_inline1.png"/> <jats:tex-math>$(X, d)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> , we investigate <jats:italic>coarse</jats:italic> flows on its uniform Roe algebra <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748025101473_inline2.png"/> <jats:tex-math>$\mathrm {C}^*_u(X)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> , defined as one-parameter groups of automorphisms whose differentiable elements include all partial isometries arising from partial translations on <jats:italic>X</jats:italic> . We first show that any flow <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748025101473_inline3.png"/> <jats:tex-math>$\sigma $</jats:tex-math> </jats:alternatives> </jats:inline-formula> on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748025101473_inline4.png"/> <jats:tex-math>$\mathrm {C}^*_u(X)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> corresponds to a (possibly unbounded) self-adjoint operator <jats:italic>h</jats:italic> on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748025101473_inline5.png"/> <jats:tex-math>$\ell _2(X)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> such that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748025101473_inline6.png"/> <jats:tex-math>$\sigma _t(a) = e^{ith} a e^{-ith}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> for all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748025101473_inline7.png"/> <jats:tex-math>$t \in \mathbb {R}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> , allowing us to focus on operators <jats:italic>h</jats:italic> that generate flows on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748025101473_inline8.png"/> <jats:tex-math>$\mathrm {C}^*_u(X)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> . </jats:p> <jats:p> Assuming Yu’s property A, we prove that a self-adjoint operator <jats:italic>h</jats:italic> on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748025101473_inline9.png"/> <jats:tex-math>$\ell _2(X)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> induces a coarse flow on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748025101473_inline10.png"/> <jats:tex-math>$\mathrm {C}^*_u(X)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> if and only if <jats:italic>h</jats:italic> can be expressed as <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748025101473_inline11.png"/> <jats:tex-math>$h = a + d$</jats:tex-math> </jats:alternatives> </jats:inline-formula> , where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748025101473_inline12.png"/> <jats:tex-math>$a \in \mathrm {C}^*_u(X)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:italic>d</jats:italic> is a diagonal operator with entries forming a coarse function on <jats:italic>X</jats:italic> . We further study cocycle equivalence and cocycle perturbations of coarse flows, showing that, under property A, any coarse flow is a cocycle perturbation of a diagonal flow. Finally, for self-adjoint operators <jats:italic>h</jats:italic> and <jats:italic>k</jats:italic> that induce coarse flows on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748025101473_inline13.png"/> <jats:tex-math>$\mathrm {C}^*_u(X)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> , we characterize conditions under which the associated flows are either cocycle perturbations of each other or cocycle conjugate to each other. In particular, if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748025101473_inline14.png"/> <jats:tex-math>$h - k$</jats:tex-math> </jats:alternatives> </jats:inline-formula> is bounded, then the flow induced by <jats:italic>h</jats:italic> is a cocycle perturbation of the flow induced by <jats:italic>k</jats:italic> . </jats:p>