Dynamical $\mathrm C^*$-algebras and coarse geometry
Bruno de Mendonça Braga
math.OA
Sep 30, 2026 · v1
math.FA
TL;DR
The paper's proofs were formalized and checked by the Lean 4 kernel using Mathlib, building on a shared formalization repository credited to collaborators.
Abstract
Both the uniform Roe algebras and the quasi-local algebras encode the large scale geometry of metric spaces. Whether these algebras coincide is a question which goes back to Roe and has only been recently solved by Ozawa. We propose a dynamical point of view on this problem. Given a set $X$, every map $h\colon X\to\mathbb{R}$ induces a one-parameter group $σ_h$ of automorphisms of $\mathcal{B}(\ell_2(X))$, given by conjugation by the diagonal unitaries $e^{ith}$, and we show that the operators which are continuity points of $σ_h$ are precisely the uniform Roe algebra of the pseudo-metric induced by $h$. If we moreover assume that $X$ is a uniformly locally finite metric space and $h$ is allowed to range over all coarse maps, this gives dynamical characterizations of both algebras: $\mathrm{C}^*_{ql}(X)$ consists exactly of the operators which are continuous for all such flows, while $\mathrm{C}^*_u(X)$ consists exactly of norm limits of those which are analytic. The regularity conditions lying between continuity and analyticity then give rise to $\mathrm{C}^*$-algebras between $\mathrm{C}^*_u(X)$ and $\mathrm{C}^*_{ql}(X)$ which are invariant under bijective coarse equivalence, and we show that this scale is not degenerate: if $X$ is a coarse disjoint union of expander graphs, the algebra generated by operators which are analytic on a strip for every diagonal flow sits strictly between $\mathrm{C}^*_u(X)$ and $\mathrm{C}^*_{ql}(X)$.
Problem
Uniform Roe algebras and quasi-local algebras both encode the large-scale geometry of metric spaces, and Ozawa showed they can differ. The paper looks for a dynamical description of both algebras and of the algebras lying between them.
Approach
Each map h: X→R induces a diagonal one-parameter automorphism group σ_h on B(ℓ2(X)), given by conjugation with e^{ith}. The paper characterizes the continuity points of σ_h as the uniform Roe algebra of the pseudo-metric induced by h. Letting h range over all coarse maps, it compares continuity and analyticity conditions and studies the asymptotics of the quasi-locality modulus. The proofs were formalized in Lean 4 with Mathlib.
Results
For u.l.f. metric spaces, the quasi-local algebra consists exactly of the operators continuous for all coarse diagonal flows. The uniform Roe algebra is the norm closure of the operators analytic for all such flows. For coarse disjoint unions of expanders, the algebra generated by operators analytic on a strip sits strictly between the two algebras.
