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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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