The coarse geometric $\ell^p$-Novikov conjecture

Let $\Gamma$ be a discrete metric space with bounded geometry. For $p\in(1,\infty)$, let $B^p(\Gamma)$ be its $\ell^p$-Roe algebra and $B_L^p(\Gamma)$ its $\ell^p$-localization algebra. For the Rips complex $P_d(\Gamma)$ at scale $d$, the evaluation homomorphism $e:B_L^p(P_d(\Gamma))\to B^p(P_d(\Gamma))$ induces the index map

$$e_*: \lim_{d\to\infty} K_*(B_L^p(P_d(\Gamma))) \to \lim_{d\to\infty} K_*(B^p(P_d(\Gamma)))\cong K_*(B^p(\Gamma)).$$

The coarse geometric $\ell^p$-Novikov conjecture. If $\Gamma$ is a discrete metric space with bounded geometry, then the index map $e_*$ is injective.

This is the coarse geometric $\ell^p$-Novikov conjecture, an $\ell^p$ analogue of the coarse Novikov conjecture. The paper proves it for metric spaces with bounded geometry that admit a coarse embedding into a simply connected complete Riemannian manifold of nonpositive sectional curvature; the general assertion remains open.