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Equivalence of Fell bundles over groups. (English) Zbl 1438.46066

The authors present a concept of equivalence for Fell bundles over groups, not necessarily saturated nor separable. They show that equivalent Fell bundles give rise to Morita-Rieffel equivalent cross-sectional \(C^*\)-algebras. They construct the so-called linking Fell bundle associated to an equivalence bundle \(\mathcal{X}\) between the Fell bundles \(\mathcal{A}\) and \(\mathcal{B}\) and then prove that there exists a \(C^*(\mathcal{A})\)-\(C^*(\mathcal{B})\)-imprimitivity bimodule \(C^*(\mathcal{X})\), which is a certain completion of \(C_c(\mathcal{X})\) such that, if \(\mathbb{L}(\mathcal{X})\) is the linking Fell bundle of \(X\), then \(C^*(\mathbb{L}(\mathcal{X})) = \mathbb{L}(C^*(\mathcal{X}))\). Further, they replace the functor \(C^*\) by another type of functors from the category of Fell bundles to the category of \(C^*\)-algebras. Such functors are a generalization of the crossed product functors studied by A. Buss et al. [Abel Symp. 12, 61–108 (2016; Zbl 1375.46048)]. Finally, they show that the equivalence of Fell bundles is transitive by defining internal tensor products of Hilbert bundles and showing that their cross-sectional algebras are isomorphic to the internal tensor product of the corresponding cross-sectional algebras of the Hilbert bundles.

MSC:

46L08 \(C^*\)-modules
46L55 Noncommutative dynamical systems

Citations:

Zbl 1375.46048
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