Projectivity, transitivity and AF-telescopes

Abstract:

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.