Ideal structure of ℓp uniform Roe algebras
Abstract
For a uniformly locally finite coarse space (X,ℰ) and p∈{0}∪[1,∞], we classify the nonzero geometric ideals of the ℓp uniform Roe algebra Bup(X,ℰ) by the ideals of ℰ, equivalently by ideals in the associated family of controlled partial coverings. Hence the geometric ideal lattice is canonically independent of p. For p∈[1,∞], we identify Bup(X,ℰ) isometrically with the reduced Lp operator algebra of the coarse groupoid and show that this identification preserves essential support, carrying geometric and ghostly ideals to dynamical and restrictive ideals. Under property A, controlled p-kernels yield a multiplier approximate identity with controlled propagation; consequently every ideal is geometric and every ghost is trivial. Thus the full ideal lattices are canonically independent of p under property A. For p∈{0,1,∞}, the complete ideal classification holds without property A; on the groupoid side, the corresponding unconditional conclusion holds for p∈{1,∞}.
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BibTeX
@misc{chung2026ideal,
title = {{Ideal structure of $\ell^p$ uniform Roe algebras}},
author = {Yeong Chyuan Chung and Xinhui Du},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.11586},
eprint = {2606.11586},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.11586},
}
RIS
TY - UNPB
TI - Ideal structure of ℓ^p uniform Roe algebras
AU - Yeong Chyuan Chung
AU - Xinhui Du
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2606.11586
UR - https://arxiv.org/abs/2606.11586
AB - For a uniformly locally finite coarse space (X,ℰ) and p∈{0}∪[1,∞], we classify the nonzero geometric ideals of the ℓ^p uniform Roe algebra B_u^p(X,ℰ) by the ideals of ℰ, equivalently by ideals in the associated family of controlled partial coverings. Hence the geometric ideal lattice is canonically independent of p. For p∈[1,∞], we identify B_u^p(X,ℰ) isometrically with the reduced L^p operator algebra of the coarse groupoid and show that this identification preserves essential support, carrying geometric and ghostly ideals to dynamical and restrictive ideals. Under property A, controlled p-kernels yield a multiplier approximate identity with controlled propagation; consequently every ideal is geometric and every ghost is trivial. Thus the full ideal lattices are canonically independent of p under property A. For p∈{0,1,∞}, the complete ideal classification holds without property A; on the groupoid side, the corresponding unconditional conclusion holds for p∈{1,∞}.
ER -