Cartan Triples and Fell Bundle Models for C*-algebras

We introduce a generalisation of Cartan pairs of C∗-algebras, which we call Cartan triples, consisting of a C∗-algebra A, a distinguished abelian subalgebra D, and the relative commutant D′ of D in A. From a Cartan triple (A,D′,D) we construct a Fell bundle E over an effective groupoid G, such that the fibres over units in the groupoid are unital C∗-algebras. We show that A is isomorphic to C∗ r (E;G), via an isomorphism which sends D to the canonical commutative subalgebra of C∗r (E;G), consisting of sections taking values in the scalar multiples of identity elements in the fibres over units. We introduce the integrated bundle of a Fell bundle B over a second countable, locally compact, Hausdorff, amenable, groupoid G with Iso◦(G) closed. The integrated bundle B is a Fell bundle over the quotient of G by Iso◦(G), which is an effective groupoid. We show that the C∗-algebra of the integrated bundle recovers the C∗-algebra of the original bundle. Furthermore, after associating a canonical Cartan triple to the C∗-algebra of a twist over an appropriate groupoid, we show that the Fell bundle constructed from this triple is isomorphic to the integrated bundle of the line bundle of the twist.