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Quantum Coarse Spaces and Quantum Uniform Roe Algebras
A noncommutative generalization of coarse geometry in which the classical relations that define a coarse structure are replaced by N. Weaver's quantum relations over an arbitrary represented von Neumann algebra M in B(H). A quantum coarse structure is a family of such quantum relations satisfying analogues of the coarse-space axioms; its quantum uniform Roe algebra is the unital C*-algebra formed as the closed union of those relations. When M is the diagonal abelian algebra l-infinity(X) in B(l2(X)), the theory reduces to classical coarse spaces and uniform Roe algebras, but M may be noncommutative or non-atomic, yielding genuinely new, non-metric examples such as support expansion C*-algebras built from a faithful normal semifinite trace. The framework also develops quantum analogues of maps between coarse spaces and their effect on the associated Roe algebras.