nLab
semiprojective morphism

Definitions

Let $A,B$ be separable $C^{*}$-algebras. A morphism $f:A\to B$ is semiprojective if for any separable $C^{*}$-algebra $C$, any increasing sequence $J_n\subset C$ of ideals with $J=\overline{{\cup }_{n=0}^{\mathrm{\infty }}{J}_n}$ and any morphism $\sigma :B\to C/J$, there exist $n$ and an ”$f$-relative lift” $\tilde{\sigma }:A\to C/J_n$ 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 $C^{*}$-algebra is semiprojective if the identity ${\mathrm{id}}_A:A\to A$ is a semiprojective morphism. In particular, every projective separable $C^{*}$-algebra is semiprojective. They are viewed as a generalization of (continuous function algebras) of ANR?s for metric spaces.

This notion is used in the strong shape theory for separable $C^{*}$-algebras.

References

• Bruce Blackadar?, Shape theory for $C^{*}$-algebras. Math. Scand. 56 (1985), no. 2, 249–275, MR87b:46074, pdf

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