## 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

Zbl 1375.46048
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