Dynamical C^*-algebras and coarse geometry

@misc{braga2026dynamical,
  title = {{Dynamical $\mathrm C^*$-algebras and coarse geometry}},
  author = {Bruno de Mendonça Braga},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39762},
  eprint = {2609.39762},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39762},
}

TY  - UNPB
TI  - Dynamical C^*-algebras and coarse geometry
AU  - Bruno de Mendonça Braga
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39762
UR  - https://arxiv.org/abs/2609.39762
AB  - 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: X→ℝ induces a one-parameter group σ_h of automorphisms of ℬ(ℓ_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: C^*_ql(X) consists exactly of the operators which are continuous for all such flows, while 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 C^*-algebras between C^*_u(X) and 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 C^*_u(X) and C^*_ql(X).
ER  -