Morita Induction and Geometric Ideals in $\ell^p$ Roe-Type Algebras

Abstract

Given two Banach algebras with bounded approximate identities that are Morita equivalent in the sense of Paravicini, we describe the induced correspondence between their closed two-sided ideals, characterize when corresponding ideals are Morita equivalent, and show that corresponding quotients are always Morita equivalent. We also prove that Morita induction preserves geometric ideals defined by compatible dense algebraic cores and give a localized submodule criterion for identifying corresponding ideals. We apply these results to the Morita equivalence between the $\ell^p$ uniform Roe algebra $B_u^p(X)$ and the $\ell^p$ uniform algebra $UB^p(X)$ of a bounded geometry metric space. As a technical ingredient, we show that $UB^p(X)$ has a bounded approximate identity of projections. Morita induction then identifies the geometric ideal lattices of the two algebras, and hence the geometric ideals of $UB^p(X)$ with the ideals of the bounded coarse structure of $X$.