nLab
semiprojective morphism

Semiprojective morphisms

Definitions

Let A,BA,B be separable C *C^\ast-algebras. A morphism f:ABf\colon A \to B is semiprojective if for any separable C *C^\ast-algebra CC, any increasing sequence J nCJ_n \subset C of ideals with J= n=0 J n¯J = \overline{\cup_{n=0}^\infty J_n} and any morphism σ:BC/J\sigma\colon B \to C/J, there exist nn and an ”ff-relative lift” σ˜:AC/J n\tilde{\sigma}\colon A \to C/J_n in the sense that the composition Aσ˜C/J nC/JA \stackrel{\tilde\sigma} \to C/J_n \to C/J equals the composition AfBσC/JA \stackrel{f}\to B \stackrel{\sigma}\to C/J, where C/J nC/JC/J_n\to C/J is the epimorphism induced by the inclusion J nJJ_n \subset J of ideals.

A separable C *C^\ast-algebra is semiprojective if the identity id A:AAid_A\colon A \to A is a semiprojective morphism. In particular, every projective separable C *C^\ast-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 *C^\ast-algebras.

References

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