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Flows on uniform Roe algebras by Braga, Bruno de Mendonça; Buss, Alcides; Exel, Ruy is a scholarly article available to read on EtoBox.

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For a uniformly locally finite metric space $(X, d)$, we investigate \emph{coarse} flows on its uniform Roe algebra $\mathrm{C}^*_u(X)$, 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 $\sigma$ on $\mathrm{C}^*_u(X)$ corresponds to a (possibly unbounded) self-adjoint operator $h$ on $\ell_2(X)$ such that $\sigma_t(a) = e^{ith} a e^{-ith}$ for all $t \in \mathbb{R}$, allowing us to focus on operators $h$ that generate flows on $ \mathrm{C}^*_u (X)$. Assuming Yu's property A, we prove that a self-adjoint operator $h$ on $\ell_2(X)$ induces a coarse flow on $\mathrm{C}^*_u(X)$ if and only if $h$ can be expressed as $h = a + d$, where $a \in \mathrm{C}^*_u(X)$ 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 $\mathrm{C}^*_u(X)$, we characterize conditions under which the associated

Author

Braga, Bruno de Mendonça; Buss, Alcides; Exel, Ruy

Published

2024

Language

EN