# Integral mappings and the principle of local reflexivity for noncommutative $L^1$-spaces

### Abstract

The operator space analogue of the strong form of the principle of local reflexivity is shown to hold for any von Neumann algebra predual, and thus for any $C^\ast$-algebraic dual. This is in striking contrast to the situation for $C^\ast$-algebras, since, for example, $K(H)$ does not have that property. The proof uses the Kaplansky density theorem together with a careful analysis of two notions of integrality for mappings of operator spaces.

Edward G. Effros

Marius Junge

Zhong-Jin Ruan