The maximal quantum group-twisted tensor product of C*-algebras (Q1747038)

If a locally compact group \(G\) acts on \(C^{*}\)-algebras \(C\) and \(D\), then the minimal and the maximal tensor products of \(C\) and \(D\) carry canonical diagonal actions of \(G\). When considering actions of \(C^{*}\)-quantum groups on \(C^{*}\)-algebras, extra structure is needed. Namely, let \(\mathbb G\) and \(\mathbb H\) be \(C^{*}\)-quantum groups in the sense of \textit{P. M. Sołtan} and \textit{S. L. Woronowicz} [J. Funct. Anal. 252, No. 1, 42--67 (2007; Zbl 1134.46044)], with algebras of functions \(A\) and \(B\) respectively. Further, one needs a bicharacter \(\chi \in \mathcal U(\widehat{A} \otimes \widehat{B})\) and two \(C^{*}\)-algebras \(C\) and \(D\) equipped with coactions of \(\mathbb G\) and \(\mathbb H\), respectively. In this setting, a quantum group-twisted tensor product of \(C\) and \(D\) was defined in [\textit{R. Meyer} et al., Int. J. Math. 25, No. 2, Article ID 1450019, 37 p. (2014; Zbl 1315.46076)] and [\textit{R. Meyer} et al., J. Noncommut. Geom. 10, No. 3, 859--888 (2016; Zbl 1362.46070)]; previously a particular case was considered in [\textit{R. Nest} and \textit{C. Voigt}, J. Funct. Anal. 258, No. 5, 1466--1503 (2010; Zbl 1191.58003)]. This construction is minimal in a suitable sense, i.e it reduces to the minimal tensor product of \(C\) and \(D\) under appropriate hypotheses. In the present paper, the authors stay in the setting described above and consider representations of \(C\) and \(D\) on the same \(C^{*}\)-algebra that commute in a braided fashion with respect to \(\chi\). To these data they attach a \(C^{*}\)-algebra \(C \boxtimes^{\chi}_{\max} D\) and a universal pair of braided-commuting representations of \(C\) and \(D\). If \(\chi\) is trivial, then it reduces to \(C \otimes_{\max} D\). It is shown that this construction shares many important properties with the mentioned above. Also \(C \boxtimes^{\chi}_{\text{max}} D\) carries an action of the a canonical coaction of the generalised Drinfeld double \(\mathcal D_{\chi}(\mathbb{G}, \mathbb{H})\), see [\textit{S. Roy}, J. Oper. Theory 74, No. 2, 485--515 (2015; Zbl 1389.81023)]. Several interesting examples illustrate this definition, notably the case when \(C\) and \(D\) are generalized Yetter-Drinfeld \(C^{*}\)-algebras. To simplify some verifications, the authors present a categorical approach where braided-commuting representations are interpreted as 2-morphisms in cubical tricategory.