Factorization through matrix spaces for finite rank operators between \(C^*\)-algebras
type: Item
instance of: scholarly article
title: Factorization through matrix spaces for finite rank operators between \(C^*\)-algebras
zbMATH Open document ID: 0947.46053
DOI: 10.1215/S0012-7094-99-10011-1
author: Marius Junge
author: Christian Le Merdy
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: The authors present factorization results involving the completely bounded norm \(\|.\|_{cb}\) and the Haagerup decomposable norm \(\|.\|_{dec}\). Here is the main theorem: Let \(A\) and \(B\) be two \(C^*\)-algebras and let \(M_n\) be the \(C^*\)-algebra of all \(n\times n\) complex matrices. Suppose \(u:A\to B\) is a finite rank bounded operator. Then: (a) For every \(\varepsilon> 0\) there exists an integer \(n\geq 1\) and two linear maps \(\alpha: A\to M_n\), \(\beta: M_n\to B\) such that \(u= \beta\alpha\) and \(\|\alpha\|_{cb}\|\beta\|_{dec}\leq (1+\varepsilon)\|u\|_{dec}\). (b) If \(A\) is a von Neumann algebra and \(u\) is \(w^*\)-continuous, then \(\alpha\) in (a) can be taken \(w^*\)-continuous, too. The paper contains also some related results and comments on the properties of the Haagerup decomposable norm.
Mathematics Subject Classification ID: 46L99
Mathematics Subject Classification ID: 47L99
Mathematics Subject Classification ID: 47A68
Mathematics Subject Classification ID: 47B49
Mathematics Subject Classification ID: 46A32
Mathematics Subject Classification ID: 46L06
zbMATH DE Number: 1425256
zbMATH Keywords: factorization results
zbMATH Keywords: completely bounded norm
zbMATH Keywords: Haagerup decomposable norm
zbMATH Keywords: finite rank bounded operator
reviewed by: Khristo N. Boyadzhiev
MaRDI profile type: Publication
cites work: The standard dual of an operator space
cites work: Tensor products of operator spaces
cites work: Q4157567
cites work: Nuclear C ∗ -Algebras and the Approximation Property
cites work: Lifting problems and local reflexivity for \(C^ *\)-algebras
cites work: Integral mappings and the principle of local reflexivity for noncommutative \(L^1\)-spaces
cites work: ON APPROXIMATION PROPERTIES FOR OPERATOR SPACES
cites work: A New Approach to Operator Spaces
cites work: The Grothendieck-Pietsch and Dvoretzky-Rogers theorems for operator spaces
cites work: Mapping Spaces and Liftings for Operator Spaces
cites work: Q3719297
cites work: Factorization of Analytic Functions with Values in Non-Commutative L1-spaces and Applications
cites work: Bounded linear operators between \(C^*\)-algebras
cites work: Bilinear forms on exact operator spaces and \(B(H) \otimes B(H)\)
cites work: C*-Nuclearity implies CPAP
cites work: Commutants of unitaries in UHF algebras and functorial properties of exactness.
cites work: On nuclear C\(^*\)-algebras
cites work: Spaces on Which each Absolutely Summing Map is Nuclear
cites work: On bounded bilinear forms
cites work: Bounded and Compact Vectorial Hankel Operators
cites work: On a quotient norm and the Sz.-Nagy-Foias lifting theorem
cites work: Every completely polynomially bounded operator is similar to a contraction
cites work: Q4722624
cites work: The maximal operator space of a normed space
cites work: Q4869299
cites work: The operator Hilbert space 𝑂𝐻, complex interpolation and tensor norms
cites work: Q4896138
cites work: Le degré de similarité d'une algèbre d'opérateurs
cites work: Q4215796
cites work: Subspaces of \(C^ *\)-algebras
cites work: Theory of operator algebras I.
cites work: Ein operatorwertiger Hahn-Banach Satz
full work available at URL: s0012-7094-99-10011-1
OpenAlex ID: W2044369950
description: scientific article; zbMATH DE number 1425256