Fell bundle C*-algebras as Cuntz–Pimsner algebras

Abstract

Consider a Fell bundle p : A → G p\colon \mathcal {A}\to G over a locally compact, Hausdorff, second countable étale groupoid G G , along with an appropriate grading c : G → Z c\colon G \to \mathbb {Z} . We prove that the reduced C ∗ \mathrm {C}^* -algebra of the Fell bundle A \mathcal {A} can be realised as the Cuntz–Pimsner algebra of a C ∗ \mathrm {C}^* -correspondence. As an application, we derive a six-term exact sequence of K \mathrm {K} -theory for C r ∗ ( G ; A ) \mathrm {C}^*_r(G; \mathcal {A}) , expressed in terms of the K \mathrm {K} -theory of C r ∗ ( G 0 ; A | G 0