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We provide a characterization of when a coarse equivalence between coarse disjoint unions of expander graphs is close to a bijective coarse equivalence. We use this to show that if the uniform Roe algebras over metric spaces that are coarse unions of expanders graphs are isomorphic, then the metric spaces must be bijectively coarsely equivalent.
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This content will become publicly available on January 1, 2026
A quantization of coarse spaces and uniform Roe algebras
We propose a quantization of coarse spaces and uniform Roe algebras. The objects are based on the quantum relations introduced by N. Weaver and require the choice of a represented von Neumann algebra. In the case of the diagonal inclusion ell_infty(X) subset B(ell_2(X)), they reduce to the usual constructions. Quantum metric spaces furnish natural examples parallel to the classical setting, but we provide other examples that are not inspired by metric considerations, including the new class of support expansion C*-algebras. We also work out the basic theory for maps between quantum coarse spaces and their consequences for quantum uniform Roe algebras.
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- Award ID(s):
- 2054860
- PAR ID:
- 10644865
- Publisher / Repository:
- MSP
- Date Published:
- Journal Name:
- Pacific Journal of Mathematics
- Volume:
- 338
- Issue:
- 1
- ISSN:
- 0030-8730
- Page Range / eLocation ID:
- 163 to 207
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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