Exotic coactions (Q2806124)

Let \(B\rtimes_{\alpha} G\) and \(B\rtimes_{\alpha,r} G\) be the full and the reduced crossed products of a \(C^*\)-dynamical system \((B, G, \alpha)\) with dual coactions \(\hat\alpha\) and \(\hat\alpha_r\), respectively. The authors give a study of ``coactions'' standing between \((B\rtimes_{\alpha} G, \hat\alpha)\) and \((B\rtimes_{\alpha,r} G, \hat\alpha_r)\) in the sense that the \(C^*\)-algebras stand between \(B\rtimes_{\alpha} G\) and \(B\rtimes_{\alpha,r} G\) in the quotient sense and the corresponding coactions are compatible with \(\hat\alpha\) and \(\hat\alpha_r\) (by abuse of notation, one uses the term coaction to denote a \(C^*\)-algebra with a coaction by \(G\)). More generally, they study coactions standing between a maximal coaction \((A,\delta)\) and its normalization \((A^n, \delta^n)\).NEWLINENEWLINESuppose that \(E\) is an ideal of the dual space of the full group \(C^*\)-algebra of \(G\) containing the dual space of the reduced group \(C^*\)-algebra. Extending the construction of \textit{N. P. Brown} and \textit{E. P. Guentner} [Bull. Lond. Math. Soc. 45, No. 6, 1181--1193 (2013; Zbl 1337.46039)], one can obtain a coaction \((A^E, \delta^E)\) standing between \((A, \delta)\) and \((A^n, \delta^n)\). The authors give a characterization of \((A^E, \delta^E)\) in terms of a property relating to \(E\).NEWLINENEWLINEFurthermore, they also provide an example of a coaction standing between \((B\rtimes_{\alpha} G, \hat\alpha)\) and \((B\rtimes_{\alpha,r} G, \hat \alpha_r)\) that does not come from this Brown-Guentner-type construction. This example is similar to the one given by \textit{A. Buss} and \textit{S. Echterhoff} [Indiana Univ. Math. J. 63, No. 6, 1659--1701 (2014; Zbl 1320.46052)].