Notes on the Lifting Theorem

We have seen that the proof of existence of inverses for elements of Ext(X) can be based on a lifting theorem for (completely) positive maps of C(X) into a quotient C∗-algebra of the form E/K, where E ⊆ B(H) is a C∗-algebra containing the compact operators K. That argument works equally well for arbitrary C∗-algebras in place of C(X) whenever a completely positive lifting exists. Thus we are led to ask if every completely positive linear map φ of an arbitrary C∗-algebra A into a quotient C∗-algebra B/K has a completely positive lifting φ0 : A → B. The answer is yes if A is nuclear by a theorem of Choi and Effros [CE76], but no in general. We will sketch a proof of the ChoiEffros theorem that is based on the existence of quasicentral approximate units; full details can be found in [Arv77]. Throughout this lecture, all Hilbert spaces are assumed to be separable.