nLab semiprojective morphism

Definitions

Let $A,B$ be separable $C^\ast$-algebras. A morphism $f\colon A \to B$ is semiprojective if for any separable $C^\ast$-algebra $C$, any increasing sequence $J_n \subset C$ of ideals with $J = \overline{\cup_{n=0}^\infty J_n}$ and any morphism $\sigma\colon B \to C/J$, there exist $n$ and an “$f$-relative lift” $\tilde{\sigma}\colon A \to C/J_n$ in the sense that the composition $A \stackrel{\tilde\sigma} \to C/J_n \to C/J$ equals the composition $A \stackrel{f}\to B \stackrel{\sigma}\to C/J$, where $C/J_n\to C/J$ is the epimorphism induced by the inclusion $J_n \subset J$ of ideals.

A separable $C^\ast$-algebra is semiprojective if the identity $id_A\colon A \to A$ is a semiprojective morphism. In particular, every projective separable $C^\ast$-algebra is semiprojective. They are viewed as a generalization of (continuous function algebras) of ANRs for metric spaces.

This notion is used in the strong shape theory for separable $C^\ast$-algebras.

References

• Bruce Blackadar, Shape theory for $C^\ast$-algebras.

  Math. Scand. 56 (1985), no. 2, 249–275, MR87b:46074, pdf

• Marius Dādārlat, Shape theory and asymptotic morphisms for $C^\ast$-algebras, Duke Math. J. 73 (3):687-711, 1994, MR95c:46117, pdf