Morita Induction and 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 of closed ideals, characterize it via localized Morita submodules, determine when corresponding ideals are Morita equivalent via restricted bimodules, and show that corresponding quotients are always Morita equivalent. We prove preservation of geometric cores associated to compatible dense algebraic cores. For equivalent étale groupoids with paracompact unit spaces, Morita induction preserves dynamical ideals and inner supports. For the Morita equivalence between the ${\ell }^p$ uniform Roe algebra and the ${\ell }^p$ uniform algebra, we prove that the latter has a bounded approximate identity of projections, identify its geometric ideals explicitly, and identify corresponding complete-lattice block decompositions of the full ideal lattices with matching extremal ideals. Under property A, or for $p=1$, these blocks collapse; for $p=2$, collapse of the minimal block characterizes property A.

arXiv comment38 pages. Substantially revised and expanded: added preservation of geometric/dynamical cores and inner supports, explicit coarse-ideal induction, complete-lattice block decompositions with extremal ideals, and new property A consequences, including a p=2 characterization via minimal-block collapse. Proofs and exposition were also streamlined