From coarse embeddability into ℓp to coarse embeddability into Hilbert space

Determine whether coarse embeddability of a discrete metric space of bounded geometry into \(\ell^p\) for a fixed \(p>2\) implies coarse embeddability into Hilbert space, without additional assumptions such as real rank zero of \(J_0(X)\) or property \((SP)\) for \(J_0(X)/K(\ell^2(X))\).

Background

The paper proves a partial result concerning the relationship between coarse embeddings into ℓp\ell^p and Hilbert space for exponents p>2p>2. Specifically, Hilbert-space coarse embeddability is equivalent to coarse embeddability into ℓp\ell^p together with either real rank zero for the ideal J0(X)J_0(X) or property (SP)(SP) for its quotient by the compact operators. The unresolved issue is whether the additional operator-algebraic hypotheses are necessary.

References

This provides a partial answer to the open question whether coarse embeddability into \ellp implies coarse embeddability into Hilbert space for a discrete metric space of bounded geometry (see ).

— Characterisations for coarse embeddability via quasi-locality and Schur multipliers  (2610.21632 - Li et al., 15 Oct 2026) in Section 1, subsection “Schur multipliers and ghost operators”