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On weakly compact operators on $$C^*$$-algebras. (English) Zbl 0593.47016
Principal theorem: Let X be a Banach space, $$X_{\infty}$$ the space of bounded functions on the ball of the dual of X; let $$J_ X:X\to X$$ be the canonical embedding: let A be a $$C^*$$-algebra and $$\Gamma_ 2(A,X_{\infty})$$ the set of operators from A to $$X_{\infty}$$ that factor through a Hilbert space. Then an operator $$T: A\to X$$ is weakly compact if and only if there is a sequence $$S_ n\in \Gamma_ 2(A,X_{\infty})$$ such that $$\lim \| J_ XT-S_ n\| =0.$$
A corollary: Every reflexive quotient of a $$C^*$$-algebra is superreflexive.
This generalizes Rosenthal’s result that every reflexive quotient of C(K) is superreflexive, being a quotient of $$L_ q(\mu)$$ for some $$q\in [2,\infty)$$ and some measure $$\mu$$ on K.
Mention is made of a recent result of Pisier’s: that every operator from a $$C^*$$-algebra A to a Banach space X of cotype q factors through a Banach space obtained by complex interpolation between A and a suitable Hilbert space. This yields a complete generalization of Rosenthal’s result.
Reviewer: P.G.Spain

##### MSC:
 47B06 Riesz operators; eigenvalue distributions; approximation numbers, $$s$$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators 47L30 Abstract operator algebras on Hilbert spaces 46L05 General theory of $$C^*$$-algebras
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