Maximal relative coarse Baum-Connes conjecture for (X, Y)

Establish that for every metric space X with bounded geometry and subspace Y ⊆ X, the maximal relative coarse assembly map μ_{max,Y,∞}: lim_{d→∞} K_*(C^*_{L,max,Y,∞}(P_d(X))) → K_*(C^*_{max,Y,∞}(X)) is an isomorphism.

Background

The maximal relative Roe algebra at infinity C*_{max,Y,∞}(X) is defined by quotienting the maximal Roe algebra by the maximal geometric ideal generated by Y. The corresponding maximal localization algebra at infinity yields a maximal relative assembly map.

The conjecture posits that this maximal assembly map is an isomorphism, enabling computation of K_(C^_{max,Y,∞}(X)) via relative K-homology at infinity and providing a maximal-norm counterpart to the reduced relative conjecture.

References

Let $X$ be a metric space with bounded geometry, $Y$ a subspace of $X$. \begin{itemize} \item The maximal relative coarse Baum-Connes conjecture for $(X,Y)$: the maximal relative coarse assembly map $\mu_{\max, Y,\infty}$ is an isomorphism; \item The maximal relative coarse Novikov conjecture for $(X,Y)$: the maximal relative coarse assembly map $\mu_{\max, Y,\infty}$ is an injection. \end{itemize}

— Relative higher index theory on quotients of Roe algebras and positive scalar curvature at infinity  (2509.23380 - Guo et al., 27 Sep 2025) in Section 3.1 (Relative coarse assembly maps)