$K$-theory of ghostly ideals for $ll^p$-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 $\ell^p$-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 ($ONL_{\mathcal P_{Fin}}$).

💡 Research Summary

The paper investigates the relationship between geometric ideals and ghostly ideals in Roe algebras of metric spaces with bounded geometry that admit a coarse embedding into an ℓ^p‑space (1 ≤ p < ∞). Ghost operators—operators whose matrix entries tend to zero at infinity but which need not be compact—form a closed ideal G, called the ghostly ideal, containing the compact ideal K. Earlier work showed that for spaces coarsely embedding into a Hilbert space, the inclusion K → G induces an isomorphism in K‑theory, which is equivalent to the coarse Baum–Connes assembly map being an isomorphism. The authors extend this phenomenon to the much broader class of ℓ^p‑embeddable spaces and, more generally, to any invariant open subset U of the Stone–Čech compactification βX.

The main theorem (Theorem 1.1) states that for any invariant open set U ⊂ βX, the canonical inclusion i : I(X,U) → G(X,U) induces an isomorphism i_* on K‑theory. Here I(X,U) is the geometric ideal generated by operators whose ε‑supports lie inside U, while G(X,U) is the corresponding ghostly ideal defined by the same support condition but without the finite‑propagation requirement. The proof proceeds in three major steps. First, the authors reduce the problem to sparse subspaces of X, because the ℓ^p‑Dirac–dual‑Dirac construction (a generalisation of the Bott‑Dirac element used in the Hilbert‑space case) is only directly applicable to such subspaces. Second, they show that on a sparse subspace the ℓ^p‑Dirac–dual‑Dirac construction restricts naturally to both I(X,U) and G(X,U), allowing a comparison inside a twisted Roe algebra. Third, within this twisted framework the distinction between geometric and ghostly elements disappears asymptotically: the ε‑supports of operators in the two ideals become indistinguishable as ε → 0, which yields the desired K‑theory isomorphism.

Several important corollaries follow. Corollary 1.2 proves the relative coarse Baum–Connes conjecture for any pair (X,Y) where X is ℓ^p‑embeddable and Y ⊂ X is arbitrary; this has implications for the existence of positive scalar curvature metrics at infinity. Corollary 1.3 establishes the operator norm localisation property for equi‑approximable finite‑rank projections (ONL_{P_{Fin}}) for ℓ^p‑embeddable spaces, showing that every ghost projection on a sparse subspace is compact. Corollary 1.4 lifts the results to the maximal setting: the canonical quotient maps from the maximal Roe algebra C^*_max(X) and from the maximal relative Roe algebras to their reduced counterparts induce K‑theory isomorphisms, and consequently the maximal relative coarse Baum–Connes conjecture holds for (X,Y).

The paper also discusses open problems. It is unknown whether the coarse groupoid G(X) of an ℓ^p‑embeddable space is K‑amenable; the authors note that their Corollary 1.4 provides indirect evidence. Further questions concern twisted versions of the coarse Baum–Connes conjecture with coefficients and the maximal conjecture for spaces admitting a fibred coarse embedding into ℓ^p‑spaces.

Methodologically, the work departs from earlier approaches that relied on a‑T‑menability and K‑amenability of the associated groupoid. Instead, it exploits the ℓ^p‑Bott‑Dirac construction and the sparsity condition to bypass the need for groupoid K‑amenability. This novel technique opens the possibility of extending similar K‑theoretic results to other Banach spaces beyond ℓ^p.

In summary, the authors prove that for any bounded‑geometry metric space coarsely embeddable into an ℓ^p‑space, the inclusion of geometric ideals into ghostly ideals is a K‑theory isomorphism for all invariant open subsets of βX. This unifies and extends previous Hilbert‑space results, yields several significant applications (relative Baum–Connes, ONL_{P_{Fin}}, maximal conjectures), and suggests new directions for research on coarse geometry, groupoid K‑theory, and Banach‑space embeddings.