\(C^*\)-algebras generated by operator systems
type: Item
instance of: scholarly article
title: \(C^*\)-algebras generated by operator systems
zbMATH Open document ID: 0940.46038
author: Eberhard Kirchberg
author: Simon Wassermann
published in: Journal of Functional Analysis
publication date: 2000-07-19T00:00:00Z
review text: First recall the definition of an operator system. Consider a complex vector space \(X\) with an involution \(x\to x^*\) and a distinguished self-adjoint element \(1\) (the unit of \(X\)). Assume that there is given a family of complete norms \(\|\cdot\|_n\) on the spaces \(X\otimes M_n(\mathbb{C})\) where \(M_n(\mathbb{C})\) is the complex vector space of \(n\times n\) matrices over \(\mathbb{C}\) and \(n= 1,2,3,\dots\)\ . Then \(X\) is called an operator system if there exists a \(C^*\)-algebra \(A\) and a self-adjoint unital embedding \(\iota: X\to A\) such that for all \(n\), the maps \(\iota\otimes id_n: X\otimes M_n(\mathbb{C})\to A\otimes M_n(\mathbb{C})\) are isometric when \(X\otimes M_n(\mathbb{C})\) is considered with the given norm \(\|\cdot\|_n\) and \(A\otimes M_n(\mathbb{C})\) is considered with the unique \(C^*\)-tensor product norm. So, roughly speaking, an operator system is a closed self-adjoint unital subspace of a \(C^*\)-algebra with unit. If \(X\) and \(Y\) are two operator systems and \(\phi\) a linear map from \(X\) to \(Y\), consider the maps \(\phi_n\) defined as \(\phi\otimes id_n\) from \(X\otimes M_n(\mathbb{C})\) to \(Y\otimes M_n(\mathbb{C})\). If \(\phi_n\) is a positive map for all \(n\), then \(\phi\) is called completely positive. If \(\phi_n\) is isometric for all \(n\), then \(\phi\) is called completely isometric. A \(C^*\)-system is an operator system \(X\) such that the double dual \(X^{**}\) is a \(C^*\)-algebra and such that the canonical embedding of \(X\) in \(X^{**}\) is a unital complete isometry. For any \(C^*\)-algebra \(A\) and any closed left ideal \(L\) of \(A\), the quotient vector space \(A/(L+ L^*)\) (where \(L^*= \{x^*\mid x\in L\}\)) carries a natural operator system structure. It is a \(C^*\)-system and one of the main results of the paper says that any separable \(C^*\)-system is of this form. It is also shown that any (inseparable) \(C^*\)-system is an inductive limit of separable \(C^*\)-systems. To any op…
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 1199813
zbMATH Keywords: operator system
zbMATH Keywords: closed self-adjoint unital subspace of a \(C^*\)-algebra with unit
zbMATH Keywords: \(C^*\)-system
zbMATH Keywords: universal \(C^*\)-algebra
zbMATH Keywords: regular \(C^*\)-algebra
zbMATH Keywords: *-homomorphism
zbMATH Keywords: nuclear
zbMATH Keywords: structure of exact \(C^*\)-algebras
reviewed by: Alfons Van Daele
MaRDI profile type: Publication
full work available at URL: jfan.1997.3226
OpenAlex ID: W2037241116
cites work: Injectivity and operator spaces
cites work: The completely positive lifting problem for \(C^*\)-algebras
cites work: On restricted perturbations in inverse images and a description of normalizer algebras in \(C^*\)-algebras
cites work: On subalgebras of the CAR-algebra
cites work: On non-semisplit extensions, tensor products and exactness of group \(C^*\)-algebras
cites work: Q4722624
cites work: Q4741226
cites work: Property T and Approximation of Operators
cites work: Tensor Products of Free-Group C* -Algebras
cites work: Subquotients of UHF C*-algebras
DOI: 10.1006/JFAN.1997.3226
description: scientific article; zbMATH DE number 1199813