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The Grothendieck-Pietsch and Dvoretzky-Rogers theorems for operator spaces. (English) Zbl 0802.46014

The factorization theorem for integral operators and an analogue of Dvoretzky-Rogers theorem is carried over in the category of operator spaces. Integral operators are defined to be the limit of completely nuclear operators in the point-norm topology. These operators essentially factorize through a non commutative “Radon-Nikodym map” \(\theta_{\theta\eta}: R\to R'_ *\) with \(\theta_{\theta \eta} (r) (r')= rr' \theta\cdot \eta\). This factorization theorem leads to a characterization of integral operators on subnuclear, i.e. exact, operator spaces.
Following Grothendieck’s concept, absolutely (matrix) summing operators are introduced, namely \(\phi: V\to W\) is absolutely summing if \(\phi\otimes id: V\otimes_ \vee T_ \infty\to W\otimes_ \wedge T_ \infty\) is continuous. Among basic properties it is proven, with the help of 2 row and 2 column summing operators, that the identity on an operator space is absolutely summing if and only if it is finite dimensional.
Reviewer: H.König (Kiel)

MSC:

46A32 Spaces of linear operators; topological tensor products; approximation properties
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