Deformation of \(C^\ast\)-algebras by cocycles on locally compact quantum groups
The authors give a method to deform \(C^*\)-algebras endowed with actions of locally compact quantum groups. The deformation data consist of a \(C^*\)-algebra \(A\) together with a (left) action of a locally compact quantum group \(G\) and a unitary \(2\)-cocycle \(\Omega\) on its dual \(\widehat{G}\). One of the main results in the paper states that under a certain regularity condition on the \(2\)-cocycle, the deformed \(C^*\)-algebra \(A_\Omega\) is stably isomorphic to a twisted bidual crossed product of the form \(\widehat{G}^{\mathrm{op}}\ltimes_\Omega G \ltimes A\). The regularity condition on \(\Omega\) means that \(C_0(G)_\Omega\ltimes G\) is canonically isomorphic to the \(C^*\)-algebra \(K(L^2(G))\) of compact operators on the Hilbert space \(L^2(G)\). This is related to regularity of \(G\) itself, but even if \(G\) is a locally compact \textit{group} it is not clear whether every \(2\)-cocycle on \(G\) is regular. In this case, the regularity is strongly related to properness and saturation (in \textit{M. A. Rieffel}'s sense [Prog. Math. 84, 141--182 (1991; Zbl 0749.22003)]) for the action of \(G\) on the deformed algebra \(C_0(G)_\Omega=C^*(\widehat{G},\Omega)\). Indeed, the authors leave this problem open and also the task to find more manageable conditions of verifying regularity in general. In the group case, the deformation process defined by the authors is related to previous deformation constructions, as studied by Rieffel, Kasprzak, \textit{P. Bieliavsky} and \textit{V. Gayral} [Mem. Am. Math. Soc. 1110 (2015; \url{doi:10.1090/memo/1115}), \url{arxiv:1109.3419}], but the exact relation is not clear in general and is also left as an open problem in the paper.
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