A Note on Approximation Properties of Crossed Products by Hilbert Bimodules

Let X be a Hilbert bimodule over a C∗-algebra A and OX = A⋊X Z. Using a finite section method we construct a sequence of completely positive contractions factoring through matrix algebras over A which act on sξs ∗ η as Schur multipliers converging to the identity. This shows immediately that for a finitely generated X the algebra OX inherits any standard approximation property such as nuclearity, exactness, CBAP or OAP from A. We discuss extensions of our techniques to general Pimsner algebras.