## Amenability for Fell bundles.(English)Zbl 0881.46046

Given a Fell bundle $$\mathcal B$$, over a discrete group $$\Gamma$$, we construct its reduced cross sectional algebra $$C^*_r({\mathcal B})$$, in analogy with the reduced crossed products defined for $$C^*$$-dynamical systems. When the reduced and full cross sectional algebras of $$\mathcal B$$ 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 $$C^*_r({\mathcal B})$$, when the bundle is amenable.
We also study induced ideals in graded $$C^*$$-algebras and obtain a generalization of results of Strătilă 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 $${\mathbf F}_n$$-grading of the Cuntz-Krieger algebras $${\mathcal O}_A$$, 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 J. Cuntz and W. Krieger in their paper [Invent. Math. 56, 251-268 (1980; Zbl 0434.46045)].

### MSC:

 46L55 Noncommutative dynamical systems 46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)

Zbl 0434.46045
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