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Actions of quantum groups on dual operator spaces and their crossed products
Abstract
We study the category of dual operator spaces equipped with an action of a locally compact quantum group 𝔾. The Fubini crossed product functor −rtimesF 𝔾 and the weak*-crossed product functor −rtimes̅𝔾 are shown to be equal if and only if 𝔾 has the approximation property of Haagerup and Kraus. Using the natural isomorphism −rtimesF𝔾≅ L1(𝔾)𝒞ℬ(B(L2(𝔾))*, −), this leads to a characterization of the approximation property of 𝔾 via an L1(𝔾)-module approximation property for B(L2(𝔾))*. Finally, exactness of the Fubini crossed product functor is investigated and related to amenability properties of 𝔾.
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BibTeX
@misc{crann2026actions,
title = {{Actions of quantum groups on dual operator spaces and their crossed products}},
author = {Jason Crann and Joeri De Ro and Jacek Krajczok},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02822},
eprint = {2609.02822},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02822},
}
RIS
TY - UNPB
TI - Actions of quantum groups on dual operator spaces and their crossed products
AU - Jason Crann
AU - Joeri De Ro
AU - Jacek Krajczok
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02822
UR - https://arxiv.org/abs/2609.02822
AB - We study the category of dual operator spaces equipped with an action of a locally compact quantum group 𝔾. The Fubini crossed product functor −rtimes^F 𝔾 and the weak^*-crossed product functor −rtimes̅𝔾 are shown to be equal if and only if 𝔾 has the approximation property of Haagerup and Kraus. Using the natural isomorphism −rtimes^F𝔾≅ _L^1(𝔾)𝒞ℬ(B(L^2(𝔾))_*, −), this leads to a characterization of the approximation property of 𝔾 via an L^1(𝔾)-module approximation property for B(L^2(𝔾))_*. Finally, exactness of the Fubini crossed product functor is investigated and related to amenability properties of 𝔾.
ER -