A Geometric Characterization of Nuclearity and Injectivity

Abstract

Let R( A , N ) be the space of bounded non-degenerate representations π: A → N , where A is a nuclear C*-algebra and N an injective von Neumann algebra with separable predual. We prove that R( A , N ) is an homogeneous reductive space under the action of the group G N , of invertible elements of N , and also an analytic submanifold of L( A , N ). The same is proved for the space of unital ultraweakly continuous bounded representations from an injective von Neumann algebra M into N . We prove also that the existence of a reductive structure for R( A , L( H)) is sufficient for A to be nuclear (and injective in the von Neumann case). Most of the known examples of Banach homogeneous reductive spaces (see [AS2], [ARS], [CPR2], [MR] and [M]) are particular cases of this construction, which moreover generalizes them, for example, to representations of amenable, type I or almost connected groups.