K-Theoretic Comparison of Roe and Quasi-Local Algebras via Projections
Abstract
A central question in higher index theory and operator algebras is whether the Roe algebra and the quasi-local algebra associated with a metric space of bounded geometry coincide, or at least have the same K-theory. In this paper, we focus on a \emph{sparse} metric space X. We prove the following three main results: (1) For a block-diagonal operator T with uniformly bounded block-rank, T is quasi-local if and only if it is in the Roe algebra. (2) In general, we discover a ghost block-diagonal projection which is quasi-local but not in the Roe algebra. (3) For a sequence of expander graphs with sufficiently large girth, the inclusion of the uniform Roe algebra into the uniform quasi-local algebra induces a \emph{non-surjective} map on their K0-groups. This yields the first known K-theoretic distinction between the uniform Roe algebra and the uniform quasi-local algebra.
Related
- Projections in reduced group C*-algebras with prescribed trace values in (0,1)
- Double cross products with projections and relative Rota-Baxter operators on Hopf algebras
- Categorical and Floer-theoretic entropies: comparison and counting conjugacy classes in symplectic mapping class groups
- Local structure of sumsets and projections of self-affine carpets
- A Cooperative Game-Theoretic Framework for Real-Time Deviation Settlement in Local Energy Communities
BibTeX
@misc{li2026ktheoretic,
title = {{$K$-Theoretic Comparison of Roe and Quasi-Local Algebras via Projections}},
author = {Kang Li and Jiawen Zhang and Jingming Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.22439},
eprint = {2608.22439},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22439},
note = {CC0},
}
RIS
TY - UNPB
TI - K-Theoretic Comparison of Roe and Quasi-Local Algebras via Projections
AU - Kang Li
AU - Jiawen Zhang
AU - Jingming Zhu
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2608.22439
UR - https://arxiv.org/abs/2608.22439
AB - A central question in higher index theory and operator algebras is whether the Roe algebra and the quasi-local algebra associated with a metric space of bounded geometry coincide, or at least have the same K-theory. In this paper, we focus on a \emph{sparse} metric space X. We prove the following three main results: (1) For a block-diagonal operator T with uniformly bounded block-rank, T is quasi-local if and only if it is in the Roe algebra. (2) In general, we discover a ghost block-diagonal projection which is quasi-local but not in the Roe algebra. (3) For a sequence of expander graphs with sufficiently large girth, the inclusion of the uniform Roe algebra into the uniform quasi-local algebra induces a \emph{non-surjective} map on their K_0-groups. This yields the first known K-theoretic distinction between the uniform Roe algebra and the uniform quasi-local algebra.
ER -