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In this paper we study deformations of $C^*$-algebras that are given as cross-sectional $C^*$-algebras of Fell bundles $\mathcal A$ over locally compact groups G. Our deformation comes from a direct deformation of the Fell bundles $\mathcal A$ via certain parameters, such as automorphisms of the Fell bundle, group cocycles, or central group extensions of G by the circle group $\mathbb T$, and then taking cross-sectional algebras of the deformed Fell bundles. We then show that this direct deformation method is equivalent to the deformation via the dual coactions by similar parameters as studied previously in [4, 7].
We give a short proof of a result of T. Bates and T. Giordano stating that any uniformly bounded Borel cocycle into a finite von Neumann algebra is cohomologous to a unitary cocycle. We also point out a separability issue in their proof. Our approach is based on the existence of a non-positive curvature metric on the positive cone of a finite von Neumann algebra.
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