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

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 ${\mathrm{\ell }}^p$-space ($1\le p<\mathrm{\infty }$), 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 ($ON{L}_{{\mathcal{P}}_{Fin}}$).