$K$-theory of ghostly ideals for $\ell^p$-coarsely embeddable spaces

TL;DR

This paper proves that for metric spaces with bounded geometry that embed into an $ extit{ ext{l}}^p$-space, the $K$-theory of geometric and ghostly ideals coincide, leading to implications for coarse Baum-Connes and localization properties.

Contribution

It establishes an isomorphism in $K$-theory between geometric and ghostly ideals for $ extit{ ext{l}}^p$-embeddable spaces, advancing coarse index theory.

Findings

Isomorphism in $K$-theory for geometric and ghostly ideals

Spaces satisfy the relative coarse Baum-Connes conjecture

Spaces have the operator norm localization property

Abstract

Ghostly ideals are among the most mysterious objects in coarse index theory. In this paper, we show that if a metric space $X$ with bounded geometry 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. As consequences, we deduce that such spaces satisfy the relative coarse Baum-Connes conjectures, as well as the operator norm localization property for finite rank projections ($ONL_{\mathcal P_{Fin}}$).