A Geometric Characterization of Nuclearity and Injectivity

✍ Scribed by E. Andruchow; G. Corach; D. Stojanoff

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 774 KB
• Volume: 133
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1995.1134

✦ Synopsis

Let (R(, \mathscr{N}, \ldots) be the space of bounded non-degenerate representations (\pi: \alpha \rightarrow, 1), where (\alpha) is a nuclear (C^{*})-algebra and, 1 an injective von Neumann algebra with separable predual. We prove that (R(\mathscr{\mathscr { C } , , 1 )}) ) is an homogeneous reductive space under the action of the group (\mathscr{G}_{1}), of invertible elements of (1^{\prime}), and also an analytic submanifold of (L(, \alpha, \ldots)). The same is proved for the space of unital ultraweakly continuous bounded representations from an injective von Neumann algebra . (|) into 1. We prove also that the existence of a reductive structure for (R(\alpha, L(H))) is sufficient for (\alpha) 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. (" 1995 Academic Press. Inc