# Semiprojective morphisms

## 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 $J=\overline{{\cup }_{n=0}^{\infty }{J}_{n}}$ and any morphism $\sigma :B\to C/J$, there exist $n$ and an ”$f$-relative lift” $\stackrel{˜}{\sigma }:A\to C/{J}_{n}$ in the sense that the composition $A\stackrel{\stackrel{˜}{\sigma }}{\to }C/{J}_{n}\to C/J$ 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

Revised on October 12, 2011 23:32:13 by Tobias Fritz (147.83.178.20)