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arXiv · 2610.34050

Quantitative $K$-homology of uniform Roe algebras

Hao Guo·Chenxi Wang·Guoliang Yu
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Abstract

We introduce quantitative $K$-homology for uniform Roe algebras of discrete proper metric spaces with bounded geometry, based on the controlled $KK$-theory of Willett and Yu. We construct a comparison map from Lipschitz $K$-theory and prove that, for chain geodesic spaces, it is an isomorphism in a quantitative sense; in particular, quantitative $K$-homology is isomorphic to the inverse limit of the Lipschitz $K$-theory groups as the Lipschitz constant tends to zero. We also establish a pairing between quantitative $K$-homology and quantitative $K$-theory.

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Hao Guo, Chenxi Wang, Guoliang Yu. 2026-10-25. Quantitative $K$-homology of uniform Roe algebras. https://arxiv.org/abs/2610.34050

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