FLOWS ON UNIFORM ROE ALGEBRAS

Speaker(s)

RUY EXEL

Affiliation

Universidade Federal de Santa Catarina, Florianópolis, Brazil

Language of the talk

English

Date

May 7, 2025, 5:15 p.m.

Information about the event

IMPAN 405 & ZOOM

Title in Polish

FLOWS ON UNIFORM ROE ALGEBRAS

Seminar

North Atlantic Noncommutative Geometry Seminar

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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 σ on C*ᵤ(X) corresponds to a (possibly unbounded) self-adjoint operator h on ℓ₂(X) such that σₜ(a) = exp(ith) a exp(-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 ℓ₂(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. This talk is based on a joint paper with Bruno Braga and Alcides Buss.

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