Morita Induction and the Preservation of Geometric and Dynamical Ideals

Abstract

Given Banach algebras with bounded approximate identities that are Morita equivalent in the sense of Paravicini, we give explicit formulas for the induced correspondence between their closed two-sided ideals, characterize when corresponding ideals are Morita equivalent through the restricted bimodules, and show that corresponding quotients are Morita equivalent. We then isolate two mechanisms for preserving distinguished classes of ideals: compatible dense algebraic cores and localized Morita submodules. The latter shows that, for equivalent locally compact Hausdorff \'etale groupoids with paracompact unit spaces, the associated Morita equivalence of reduced groupoid $L^p$-operator algebras preserves dynamical ideals. The former applies to the Morita equivalence between the $\ell^p$ uniform Roe algebra and the $\ell^p$ uniform algebra of a bounded geometry metric space, identifying their geometric ideals. As a technical ingredient, we prove that the $\ell^p$ uniform algebra has a bounded approximate identity of projections.