K-theory for commutants in the Calkin algebra

Let A be a separable unital C*-algebra, and let B c be the commutant in the Calkin algebra of the image B of A under a trivial extension.We show that K O (B C ) is isomorphic to the group of invertibles in (weak) Ext of A and that, in the presence of an appropriate homotopy invariance assumption, K^B 0 ) is isomorphic to Ext of the reduced suspension of A. These facts lead to an alternative approach to the Pimsner-Voiculescu exact sequence for Ext of a crossed product.