×

Integral mappings and the principle of local reflexivity for noncommutative \(L^1\)-spaces. (English) Zbl 0957.47051

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 \(F \otimes V^{**}\) coincides with \((F \otimes V)^{**}\) for all finite-dimensional operator spaces \(F\). The authors go on to show that preduals of von Neumann algebras are locally reflexive, and they use an approximation property of mappings from the dual of a \(C^*\)-algebra to another \(C^*\)-algebra to deduce that such a predual \(V\) is even strongly locally reflexive in the sense that for every finite-dimensional subspace \(F\) of \(V^{**}\), every finite-dimensional subspace \(N\) of \(V^{*}\) and every \(\varepsilon>0\) there exists a mapping \(\varphi: F\to V\) such that \(\|\varphi \|_{cb} \|\varphi^{-1} \|_{cb} \leq 1+\varepsilon\), \(\langle \varphi(v),f \rangle = \langle v,f \rangle\) on \(F\times N\) and \(\varphi (v)=v\) on \(F\cap V\); this is the completely bounded analogue of the principle of local reflexivity for Banach spaces. One should note, however, that in the setting of \(C^*\)-algebras rather than their (pre-) duals, all exact \(C^*\)-algebras are locally reflexive, but the full group \(C^*\)-algebra \(C^*({\mathbb F}_{2})\) is not, and only very few \(C^*\)-algebras are strongly locally reflexive, for instance \(K(H)\) and \(B(H)\) are not.

MSC:

47L25 Operator spaces (= matricially normed spaces)
46L07 Operator spaces and completely bounded maps
46B07 Local theory of Banach spaces
46B08 Ultraproduct techniques in Banach space theory