Amenability for Fell bundles.

*(English)*Zbl 0881.46046Given 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)].

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)].

Reviewer: R.Exel (São Paulo)

##### MSC:

46L55 | Noncommutative dynamical systems |

46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |