On the Local Lifting Property for Operator spaces
by Seung-Hyeok Kye and Zhong-Jin Ruan
J. Funct. Anal., 168 (1999), 355-379

We study the local lifting property for operator spaces. This is a natural non-commutative analogue of the Banach space local lifting property, but is very much different from the local lifting property studied in $C^*$-algebra theory. We show that an operator space has the $\lambda$-local lifting property if and only if it is an ${\mathcal L\Gamma}_{1, \lambda}$ space. These operator spaces are $\lambda$-completely isomorphic to the operator subspaces of the operator preduals of von Neumann algebras, and thus $\lambda$-locally reflexive. Moreover, we show that an operator space $V$ has $\lambda$-local lifting property if and only if its operator space dual $V^*$ is $\lambda$-injective.