Xinhui Du | Searcharxiv

arXiv2026

Morita Induction and the Preservation of Geometric and Dynamical Ideals

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.

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arXiv2026

Ideal structure of $\ell^p$ uniform Roe algebras

For a uniformly locally finite coarse space $(X,\mathcal{E})$, we prove that for every $p\in\{0\}\cup[1,\infty]$, the lattice of geometric ideals in the $\ell^p$ uniform Roe algebra $B^p_u(X,\mathcal{E})$ is isomorphic to the lattice of ideals of $\mathcal{E}$ (equivalently, to the lattice of ideals in the associated family of controlled partial coverings of $X$). In particular, the lattices of geometric ideals for different values of $p$ coincide. Using limit operators, we establish a canonical isometric isomorphism between $B^p_u(X,\mathcal{E})$ and the reduced $L^p$ operator algebra of the coarse groupoid for $p\in[1,\infty]$, and show that it induces an isomorphism between lattices of ideals that preserves inner support. In particular, geometric (resp. ghostly) ideals correspond precisely to dynamical (resp. restrictive) ideals under this isomorphism. Using equivalent formulations of property A for coarse spaces, we prove that for $p\in(1,\infty)$, property A implies that $B^p_u(X,\mathcal{E})$ admits a multiplier approximate identity with controlled propagation, that all ideals are geometric, and that all ghosts are trivial. For the extreme cases $p\in\{0,1,\infty\}$, these properties hold for every uniformly locally finite coarse space without assuming Property A. Finally, for $p\in[1,\infty)$, a Morita equivalence between the $\ell^p$ uniform Roe algebra and the $\ell^p$ uniform algebra is shown to preserve the lattice of geometric ideals.

math.FA↗