Summary: Continuing our study of projective \(C^{*}\)-algebras, we establish a projective transitivity theorem generalizing the classical Glimm-Kadison result. This leads to a short proof of Glimm’s theorem that every \(C^{*}\)-algebra not of type I contains a \(C^{*}\)-subalgebra which has the Fermion algebra as a quotient. Moreover, we are able to identify this subalgebra as a generalized mapping telescope over the Fermion algebra. We next prove what we call the multiplier realization theorem. This is a technical result, relating projective subalgebras of a multiplier algebra \(M(A)\) to subalgebras of \(M(E)\), whenever \(A\) is a \(C^{*}\)-subalgebra of the corona algebra \(C(E)=M(E)/E\). We developed this to obtain a closure theorem for projective \(C^{*}\)-algebras, but it has other consequences, one of which is that if \(A\) is an extension of an MF (matricial field) algebra (in the sense of Blackadar and Kirchberg) by a projective \(C^{*}\)-algebra, then \(A\) is MF.
The last part of the paper contains a proof of the projectivity of the mapping telescope over any AF (inductive limit of finite-dimensional) \(C^{*}\)-algebra. Translated to generators, this says that in some cases it is possible to lift an infinite sequence of elements, satisfying infinitely many relations, from a quotient of any \(C^{*}\)-algebra.