Lifting problems and local reflexivity for \(C^ *\)-algebras
type: Item
instance of: scholarly article
title: Lifting problems and local reflexivity for \(C^ *\)-algebras
zbMATH Open document ID: 0613.46047
DOI: 10.1215/S0012-7094-85-05207-X
author: Edward G. Effros
author: Uffe Haagerup
published in: Duke Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: In [Ann. Math., II. Ser. 104, 585-609 (1976; Zbl 0361.46067)], \textit{M.-D. Choi} and \textit{E. G. Effros} proved that if A is an operator system (a unital linear self-adjoint subspace of operators on a Hilbert space), B is a unital \(C^*\)-algebra and J is a closed two-sided ideal in B, then one obstruction to the existence of a completely positive lifting of a completely positive map \(\phi\) :A\(\to B/J\) is the irregular behavior of minimal tensor products. Specifically, the obstruction is caused by the fact that the kernel of the map \(A\otimes_{\min}B\to A\otimes_{\min}(B/J)\) may not equal \(A\otimes_{\min}J\). In the paper under review, the authors prove that the above obstruction is the only one, by proving the following theorem: Suppose that \(J\) is a nuclear ideal in a unital \(C^*\)-algebra \(B\). Then the following are equivalent: (1) One can always lift a completely positive unital map \(\phi\) :A\(\to B/J\) to a completely positive unital map \(\psi:A\to B\) for any separable unital \(C^*\)-algebra \(A\). (2) The kernel of \(B\otimes_{\min}C\to (B/J)\otimes_{\min}C\) is \(J\otimes_{\min}C\) for all unital \(C^*\)-algebras \(C\). A \(C^*\)-algebra \(A\) is said to be approximately injective if given finite-dimensional operator systems \(E_ 1\subseteq E_ 2\subseteq {\mathcal B}({\mathcal H})\), any completely positive map \(\phi_ 1:E_ 1\to A\) has completely positive approximate extensions \(\phi_ 2:E_ 2\to A\), meaning that given \(\epsilon >0\), there exists a completely positive \(\phi_ 2:E_ 2\to A\) such that \(\| \phi_ 2|_{E_ 1}- \phi_ 1\| <\epsilon\). The theorem above is actually proven under the weaker hypothesis that J is approximately injective, although it is unknown whether any hypothesis on J is necessay. The authors then consider the property of approximate injectivity, prove that it is weaker than nuclearity or (trivially) injectivity, and that it is preserved under direct limits. Finally, the authors consider a condition on a \(C^*\)-…
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46A22
zbMATH DE Number: 3992191
zbMATH Keywords: operator system
zbMATH Keywords: closed two-sided ideal
zbMATH Keywords: existence of a completely positive lifting of a completely positive map
zbMATH Keywords: minimal tensor products
zbMATH Keywords: nuclear ideal in a unital \(C^*\)-algebra
zbMATH Keywords: approximately injective
zbMATH Keywords: completely positive approximate extensions
zbMATH Keywords: local reflexivity
MaRDI profile type: Publication
cites work: A \(C^*\)-algebra \(\mathcal A\) for which Ext (\(\mathcal A\)) is not a group
cites work: C* -Tensor Norms and Slice Maps
cites work: Subalgebras of \(C^ *\)-algebras
cites work: Q4069313
cites work: Notes on extensions of \(C^*\)-algebras
cites work: Representation and index theory for \(C^*\)-algebras generated by commuting isometries
cites work: Multipliers of the Fourier Algebras of Some Simple Lie Groups and Their Discrete Subgroups
cites work: A Simple C*-Algebra Generated by Two Finite-Order Unitaries
cites work: Injectivity and operator spaces
cites work: Nuclear C ∗ -Algebras and the Approximation Property
cites work: Q4157567
cites work: The completely positive lifting problem for \(C^*\)-algebras
cites work: Lifting Problems and the Cohomology of C*-Algebras
cites work: The Equation L(E, X ∗ ∗ = L(E, X) ∗ ∗ and the Principle of Local Reflexivity
cites work: On the C ∗ -Algebra of Toeplitz Operators on the Quarter-Plane
cites work: Q3254057
cites work: Q4180803
cites work: Tensor products of operator algebras
cites work: Q5635882
cites work: On nuclear C\(^*\)-algebras
cites work: Q4136031
cites work: Completely Bounded Maps on C ∗ -Algebras and Invariant Operator Ranges
cites work: On tensor products of certain group \(C^*\)-algebras
cites work: Liftings in C* -Algebras: a Counterexample
cites work: Ein operatorwertiger Hahn-Banach Satz
full work available at URL: s0012-7094-85-05207-x
OpenAlex ID: W1979779217
description: scientific article; zbMATH DE number 3992191