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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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This content will become publicly available on October 1, 2026
Support expansion C*-algebras
We consider operators on L_2 spaces that expand the support of vectors in a manner controlled by some constraint function. The primary objects of study are C*-algebras that arise from suitable families of constraints, which we call support expansion C*-algebras. In the discrete setting, support expansion C*-algebras are classical uniform Roe algebras, and the continuous version featured here provides examples of “measurable" or “quantum" uniform Roe algebras as developed in a companion paper. We find that in contrast to the discrete setting, the poset of support expansion C*-algebras inside B(L_2(R)) is extremely rich, with uncountable ascending chains, descending chains, and antichains.
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- Award ID(s):
- 2054860
- PAR ID:
- 10644864
- Publisher / Repository:
- Theta Foundation
- Date Published:
- Journal Name:
- Journal of operator theory
- Volume:
- 92
- Issue:
- 2
- ISSN:
- 0379-4024
- Page Range / eLocation ID:
- 463--504
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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