On weakly compact operators on \(C^*\)-algebras
type: Item
instance of: scholarly article
title: On weakly compact operators on \(C^*\)-algebras
zbMATH Open document ID: 0593.47016
DOI: 10.1007/BF01451412
author: Hans Jarchow
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164060
review text: 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.
Mathematics Subject Classification ID: 47B06
Mathematics Subject Classification ID: 47L30
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 3953613
zbMATH Keywords: canonical embedding
zbMATH Keywords: \(C^*\)-algebra
zbMATH Keywords: Every reflexive quotient of a \(C^*\)-algebra is superreflexive
zbMATH Keywords: cotype q
zbMATH Keywords: complex interpolation
MaRDI profile type: Publication
cites work: The Dual Space of an Operator Algebra
cites work: Weak compactness in the dual space of a \(C^*\)-algebra
cites work: Opérateurs uniformément convexifiants
cites work: Q3918541
cites work: Q3721255
cites work: On subspaces of L\(^p\)
cites work: Q3876492
description: scientific article; zbMATH DE number 3953613