On the Local Lifting Property for Operator Spaces

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 different from the local lifting property studied in C*-algebra theory. We show that an operator space has the λ-local lifting property if and only if it is an LΓ1, λ space. These operator space are λ-completely isomorphic to the operator subspaces of the operator preduals of von Neumann algebras, and thus λ-locally reflexive. Moreover, we show that an operator space V has the λ-local lifting property if and only if its operator space dual V* is λ-injective.