Kirchberg's conjecture in the system of Hilbert space factorable mapping spaces

This study aims to introduce the notions of injectivity, local reflexivity, exactness, and nuclearity in the system ((Gamma 2c(.,.),gamma 2c(.))\We find that every dual operator space is injective in the system ((Gamma 2c(.,.),gamma 2c(.))and nuclearity is equivalent to exactness in this system. As a corollary, we prove that Kirchberg's conjecture on the equivalence of exactness and local reflexivity forC* -algebras is false in this system, i.e., there exists aC*-algebraAthat is locally reflexive in this system but is not exact in this system.