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arXiv · 2609.06305

The K-theory of uniform Roe algebras for coarse structures generated by finite-rank free abelian subgroups

Charles Fanning·Mehmet Emin Aktas
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Abstract

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 $\ell^2(X)$. The $K$-theory of uniform Roe algebras is known in asymptotic dimension zero, but it is not fully understood in higher dimensions. We compute $K_0(C_u^*(G,\mathcal E))$ and $K_1(C_u^*(G,\mathcal E))$ for every countable discrete abelian group $G$ and every finite-rank free abelian subgroup $H\leq G$, where $\mathcal E$ is the coarse structure generated by $H$. We use the Proietti--Yamashita spectral sequence to express the $K$-theory in terms of $H_*(H;\ell^\infty(G,\mathbb Z))$, which we then compute.

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Charles Fanning, Mehmet Emin Aktas. 2026-09-05. The K-theory of uniform Roe algebras for coarse structures generated by finite-rank free abelian subgroups. https://arxiv.org/abs/2609.06305

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