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Isomorphisms of ℓ1-crossed products of topological systems
Abstract
We give a complete description of the isometric isomorphisms between the ℓ1-crossed products of actions of discrete groups on nonempty compact Hausdorff spaces. In particular, two such crossed products are isometrically isomorphic as Banach algebras if and only if the actions are topologically isomorphic. It follows that ℓ1(Gcurvearrowright X) remembers not just X but even the acting group and the action, and is not determined by the transformation groupoid Gltimes X alone. The proof recovers C(X) from the hermitian elements and uses the geometry of Lp-spaces to identify the invertible isometries. We also obtain analogous results for measurable actions.
Related
- Selfless nonnuclear crossed products
- Simplicity of reduced crossed products
- KS-Groupoids and KS-Crossed Products associated with Inverse Semigroup Actions
- Invariant von Neumann subalgebras of crossed products and tensor products
- Actions of quantum groups on dual operator spaces and their crossed products
BibTeX
@misc{gardella2026isomorphisms,
title = {{Isomorphisms of $\ell^1$-crossed products of topological systems}},
author = {Eusebio Gardella and Tabaré Roland},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33050},
eprint = {2610.33050},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33050},
}
RIS
TY - UNPB
TI - Isomorphisms of ℓ^1-crossed products of topological systems
AU - Eusebio Gardella
AU - Tabaré Roland
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.33050
UR - https://arxiv.org/abs/2610.33050
AB - We give a complete description of the isometric isomorphisms between the ℓ^1-crossed products of actions of discrete groups on nonempty compact Hausdorff spaces. In particular, two such crossed products are isometrically isomorphic as Banach algebras if and only if the actions are topologically isomorphic. It follows that ℓ^1(Gcurvearrowright X) remembers not just X but even the acting group and the action, and is not determined by the transformation groupoid Gltimes X alone. The proof recovers C(X) from the hermitian elements and uses the geometry of L^p-spaces to identify the invertible isometries. We also obtain analogous results for measurable actions.
ER -