$K$-theory of ghostly ideals for $\ell^p$-coarsely embeddable spaces
Liang Guo, Kang Li, Qin Wang
TL;DR
The paper proves that for metric spaces with bounded geometry which coarsely embed into an $\ell^p$-space, the inclusion from geometric to ghostly ideals in the Roe algebra induces a $K$-theory isomorphism for any invariant open set. The authors replace groupoid techniques with a twisted Roe algebra framework built from an $\ell^p$ Bott–Dirac construction, establishing a Dirac–dual–Dirac reduction to compare twisted geometric and ghostly ideals. This yields corollaries including the relative and maximal coarse Baum–Connes conjectures and operator norm localization (ONL) for equi-approximable finite-rank projections, broadening the scope beyond Hilbert-space embeddability. The work provides new tools for K-theory computations in coarse geometry and advances understanding of how $\ell^p$-geometry interacts with index theory via twisted operator algebras.
Abstract
Let $X$ be a metric space with bounded geometry. We show that if $X$ admits a coarse embedding into an $\ell^p$-space ($1 \le p < \infty$), then the canonical inclusion from any geometric ideal to the corresponding ghostly ideal induces an isomorphism in $K$-theory. Our approach relies on the construction of twisted Roe algebras, avoiding the use of groupoid techniques. As consequences, we deduce the relative (maximal) coarse Baum-Connes conjectures for such spaces, as well as the Operator Norm Localization property for finite rank projections ($ONL_{\mathcal{P}_{Fin}}$).