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 ℓp uniform Roe algebra and the ℓp 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.
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BibTeX
@misc{chung2026morita,
title = {{Morita Induction and Preservation of Geometric and Dynamical Ideals}},
author = {Yeong Chyuan Chung and Xinhui Du},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.30380},
eprint = {2608.30380},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30380},
}
RIS
TY - UNPB
TI - Morita Induction and Preservation of Geometric and Dynamical Ideals
AU - Yeong Chyuan Chung
AU - Xinhui Du
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2608.30380
UR - https://arxiv.org/abs/2608.30380
AB - 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 ℓ^p uniform Roe algebra and the ℓ^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.
ER -