Morita equivalence for L^p-operator algebras associated with étale groupoids

Abstract

We prove that equivalent locally compact, locally Hausdorff, étale groupoids with paracompact unit spaces have Morita equivalent reduced and full L^p-operator algebras for every p∈[1,∞]. The reduced theorem requires pairing-valued approximate identities, reflecting the failure of unconditionality of the reduced norm for 1<p<∞, while the full theorem rests on a full-clopen reduction theorem proved by dilating spatial representations. For 1≤ p<∞, we also compare the concrete p-linking algebra of the reduced Morita equivalence with the reduced L^p-operator algebra of the linking groupoid: the canonical comparison is contractive, injective, and has dense range in general, and is an isometric isomorphism for p=2. We further study Morita cycles arising from proper étale groupoid correspondences, including the reduced case under suitable extension hypotheses. Applications include an L^p version of Green's symmetric imprimitivity theorem, results for coarse groupoids and inverse semigroups, and invariance of the corresponding K-theory.

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BibTeX RIS

BibTeX

@misc{chung2026morita,
  title = {{Morita equivalence for $L^p$-operator algebras associated with étale groupoids}},
  author = {Yeong Chyuan Chung and Alonso Delfín and Zhen Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39031},
  eprint = {2609.39031},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39031},
}

RIS

TY  - UNPB
TI  - Morita equivalence for L^p-operator algebras associated with étale groupoids
AU  - Yeong Chyuan Chung
AU  - Alonso Delfín
AU  - Zhen Wang
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39031
UR  - https://arxiv.org/abs/2609.39031
AB  - We prove that equivalent locally compact, locally Hausdorff, étale groupoids with paracompact unit spaces have Morita equivalent reduced and full L^p-operator algebras for every p∈[1,∞]. The reduced theorem requires pairing-valued approximate identities, reflecting the failure of unconditionality of the reduced norm for 1<p<∞, while the full theorem rests on a full-clopen reduction theorem proved by dilating spatial representations. For 1≤ p<∞, we also compare the concrete p-linking algebra of the reduced Morita equivalence with the reduced L^p-operator algebra of the linking groupoid: the canonical comparison is contractive, injective, and has dense range in general, and is an isometric isomorphism for p=2. We further study Morita cycles arising from proper étale groupoid correspondences, including the reduced case under suitable extension hypotheses. Applications include an L^p version of Green's symmetric imprimitivity theorem, results for coarse groupoids and inverse semigroups, and invariance of the corresponding K-theory.
ER  -

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