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

Edward G. Effros, Marius Junge and Zhong-Jin Ruan

Review from Zentralblatt MATH:

In the theory of operator spaces, i.e., subspaces $V$ of $B(H)$ endowed with the matricial norms on $M_{n}(V)$ inherited from $B(H^n)$, one studies concepts of functional analysis from a ``complete'' point of view, often encountering unexpected phenomena. One such phenomenon is that the principle of local reflexivity, one of the most versatile tools in Banach space theory, generally fails in the category of operator spaces.

The main result of this paper is that preduals of von Neumann algebras, however, always satisfy a strong version of the principle of local reflexivity for operator spaces. To prove this result the authors first perform a careful study of the relationships between the classes of completely nuclear, completely integral and exactly integral mappings between operator spaces. The latter class, introduced in M. Junge's Habilitationsschrift, is somewhat larger than the class of completely integral mappings. If $\varphi: V\to W$ is completely integral, then its adjoint $\varphi^*$ need not be completely integral, but is only exactly integral. However, $\varphi^*$ is always completely integral with the same integral norm as $\varphi$ if and only if $V$ is locally reflexive. Here $V$ is called locally reflexive if the injective operator space tensor product $F \otimes V^{

Reviewed by Dirk Werner

Keywords: operator spaces; preduals of von Neumann algebras; local reflexivity; completely nuclear maps; completely integral maps; exactly integral maps; matricial norms; injective operator space tensor product; exact $C^*$-algebras

Classification (MSC2000): 47L25 46L07 46B07 46B08

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