A Geometric Characterization of Nuclearity and Injectivity

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 GN, 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.