Amenability for Fell Bundles

Given a Fell bundle $\B$, over a discrete group $\Gamma$, we construct its reduced cross sectional algebra $C^*_r(\B)$, in analogy with the reduced crossed products defined for C*-dynamical systems. When the reduced and full cross sectional algebras of $\B$ are isomorphic, we say that the bundle is amenable. We then formulate an approximation property which we prove to be a sufficient condition for amenability. A theory of $\Gamma$-graded C*-algebras possessing a conditional expectation is developed, with an eye on the Fell bundle that one naturally associates to the grading. We show, for instance, that all such algebras are isomorphic to $C^*_r(\B)$, when the bundle is amenable. We also study induced ideals in graded C*-algebras and obtain a generalization of results of Stratila and Voiculescu on AF-algebras, and of Nica on quasi-lattice ordered groups. A brief comment is made on the relevance, to our theory, of a certain open problem in the theory of exact C*-algebras. An application is given to the case of an $F_n$-grading of the Cuntz-Krieger algebras $O_A$, recently discovered by Quigg and Raeburn. Specifically, we show that the Cuntz-Krieger bundle satisfies the approximation property, and hence is amenable, for all matrices $A$ with entries in {0,1}, even if $A$ does not satisfy the well known property (I) studied by Cuntz and Krieger in their paper.