A GEOMETRIC CHARACTERIZATION OF NUCLEARITY AND INJECTIVITY

Let R(A,N) be the space of bounded non-degenerate representations pi: A --> N, where A is a nuclear C--algebra and N an injective von Neuma nn algebra with separable predual. We prove that R(A,N) is an homogene ous 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 represe ntations from an injective von Neumann algebra. M into N. We prove als o that the existence of a reductive structure for R(A,L(H)) is suffici ent 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 construct ion, which moreover generalizes them, for example, to representations of amenable, type I or almost connected groups. (C) 1995 Academic Pres s. Inc.