Inductive limits of projective $C$*-algebras

Abstract

We show that a separable $C$*-algebra is an inductive limits of projective $C$*-algebras if and only if it has trivial shape, that is, if it is shape equivalent to the zero $C$*-algebra. In particular, every contractible $C$*-algebra is an inductive limit of projectives, and one may assume that the connecting morphisms are surjective. Interestingly, an example of Dadarlat shows that trivial shape does not pass to full hereditary sub-$C$*-algebras. It then follows that the same fails for projectivity.

To obtain these results, we develop criteria for inductive limit decompositions, and we discuss the relation with different concepts of approximation.

As a main application of our findings we show that a $C$-algebra is (weakly) projective if and only if it is (weakly) semiprojective and has trivial shape. It follows that a $C$-algebra is projective if and only if it is contractible and semiprojective. This confirms a conjecture of Loring.