Decomposable approximations revisited

Nuclear C*-algebras enjoy a number of approximation properties, most famously the completely positive approximation property. This was recently sharpened to arrange for the incoming maps to be sums of order-zero maps. We show that, in addition, the outgoing maps can be chosen to be asymptotically order-zero. Further these maps can be chosen to be asymptotically multiplicative if and only if the C*-algebra and all its traces are quasidiagonal.
New section 4 added, providing a lifting lemma needed for the statement of Prop 3.2. Footnote 5 added to end of proof of Prop 3.2, and bibliography and precise locations of references updated. No other changes made to sects 1-3. 19 Pages