Crossed products and their finite dimensional approximation properties.

摘要

We prove that the crossed product A x G of a unital separable quasidiagonal C*-algebra A by a discrete countable amenable maximally almost periodic group G is quasidiagonal, provided that the action is almost periodic. This generalizes a result of M. Pimsner and D. Voiculescu. As an application we consider a broad family of crossed products of the form C( G˜) x G, which includes the Bunce--Deddens algebras, for which we prove quasidiagonality using our result.;The above-mentioned crossed products, which we call generalized Bunce--Deddens algebras, turn out to have many desirable properties. Apart from quasidiagonal, we show that they are unital separable simple nuclear algebras, of real rank zero, stable rank one, with comparability of projections and a unique trace. Half of these results come via the almost AF groupoids approach, due to N. C. Phillips. Finally, we formulate an open problem concerning the tracial rank of the generalized Bunce--Deddens algebras, which could lead to their classification by their ordered K-theory, as AH algebras of no dimension growth.