Weak Semiprojectivity in Purely Infinite Simple C *-Algebras

Source abstract

Abstract Let A be a separable amenable purely infinite simple C *-algebra which satisfies the Universal Coefficient Theorem. We prove that A is weakly semiprojective if and only if K i ( A ) is a countable direct sum of finitely generated groups ( i = 0, 1). Therefore, if A is such a C *-algebra, for any ε > 0 and any finite subset ℱ ⊂ A there exist δ > 0 and a finite subset ⊂ A satisfying the following: for any contractive positive linear map L : A → B (for any C *-algebra B ) with ∥ L ( ab ) – L ( a ) L ( b )∥ < δ for a , b ∈ there exists a homomorphism h : A → B such that ∥ h ( a ) – L ( a )∥ < ε for a ∈ ℱ.