PreprintarXiv licencearXiv
The K-theory of uniform Roe algebras for coarse structures generated by finite-rank free abelian subgroups
Abstract
For a uniformly locally finite coarse space X, the uniform Roe algebra Cu*(X) is the operator norm closure of the controlled operators on ℓ2(X). The K-theory of uniform Roe algebras is known in asymptotic dimension zero, but it is not fully understood in higher dimensions. We compute K0(Cu*(G,ℰ)) and K1(Cu*(G,ℰ)) for every countable discrete abelian group G and every finite-rank free abelian subgroup H≤ G, where ℰ is the coarse structure generated by H. We use the Proietti--Yamashita spectral sequence to express the K-theory in terms of H*(H;ℓ∞(G,ℤ)), which we then compute.
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BibTeX
@misc{fanning2026ktheory,
title = {{The K-theory of uniform Roe algebras for coarse structures generated by finite-rank free abelian subgroups}},
author = {Charles Fanning and Mehmet Emin Aktas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06305},
eprint = {2609.06305},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06305},
}
RIS
TY - UNPB
TI - The K-theory of uniform Roe algebras for coarse structures generated by finite-rank free abelian subgroups
AU - Charles Fanning
AU - Mehmet Emin Aktas
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06305
UR - https://arxiv.org/abs/2609.06305
AB - For a uniformly locally finite coarse space X, the uniform Roe algebra C_u^*(X) is the operator norm closure of the controlled operators on ℓ^2(X). The K-theory of uniform Roe algebras is known in asymptotic dimension zero, but it is not fully understood in higher dimensions. We compute K_0(C_u^*(G,ℰ)) and K_1(C_u^*(G,ℰ)) for every countable discrete abelian group G and every finite-rank free abelian subgroup H≤ G, where ℰ is the coarse structure generated by H. We use the Proietti--Yamashita spectral sequence to express the K-theory in terms of H_*(H;ℓ^∞(G,ℤ)), which we then compute.
ER -