The lifting property for $C^{\ast }$-algebras : from local to global ?

This note is motivated by Kirchberg’s conjecture that the local lifting property (LLP) implies the lifting property (LP) for $C^{\ast }$-algebras. The author recently constructed by a “local” method an example of $C^{\ast }$-algebra with the LLP and the weak expectation property (WEP) which might be a counterexample. We give here several equivalent conditions all equivalent to the validity of the implication LLP $\Rightarrow$ LP for WEP $C^{\ast }$-algebras, or equivalently for quotients of WEP $C^{\ast }$-algebras (QWEP), that hopefully clarify the nature of the problem. But unfortunately we cannot decide whether they hold true. These conditions highlight the notion of “controlable” finite dimensional (f.d.) operator space, in connection with certain $C^{\ast }$-tensor products. The latter led us to a closely related variant of local reflexivity.