On the local lifting property for operator spaces

We study the local lifting property for operator spaces. This is a natural noncommutative 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 lambda-local lifting property if a nd only if it is an L Gamma(1,lambda) space. These operator space are lambd a-completely isomorphic to the operator subspaces of the operator preduals of von Neumann algebras, and thus lambda-locally reflexive. Moreover, we sh ow that an operator space V has the lambda-local lifting property if and on ly if its operator space dual V* is lambda-injective. (C) 1999 Academic Pre ss.