Matrix convexity: Operator analogues of the bipolar and Hahn-Banach theorems
type: Item
instance of: scholarly article
title: Matrix convexity: Operator analogues of the bipolar and Hahn-Banach theorems
zbMATH Open document ID: 0897.46046
author: Edward G. Effros
author: Soren Winkler
published in: Journal of Functional Analysis
publication date: 1998-10-19T00:00:00Z
review text: Let \(V\) and \(V'\) be dual vector spaces (over \(\mathbb{C}\)). This means that there is a non-degenerate pairing between the two spaces so that each of them can be seen as a subspace of the dual space of the other. Associated with the pairing come weak topologies on these vector spaces. If \(K\) is a convex subset of \(V\), the polar set \(K^0\) in \(V'\) is defined as \[ K^0= \{f\in V'\mid\text{Re}\langle v, f\rangle\leq 1\text{ for all }v\in K\}. \] It is a weakly closed convex set in \(V'\), containing \(0\). The classical bipolar theorem says that \(K^{00}= K\) if \(0\in K\) and \(K\) is weakly closed to begin with. The result is closely related to, and usually proved by means of the algebraic form of the Hahn-Banach theorem. This paper is a contribution to the further development of `quantized functional analysis'. It treats the quantized forms of the bipolar theorem and the Hahn-Banach theorem. The basic notion in this context is a matrix convex set, already introduced by \textit{G. Wittstock} in [North Holland Math. Stud. 90, 175-188 (1984; Zbl 0589.46003)]. As before, let \(V\) be a vector space (over \(\mathbb{C}\)) and consider the vector spaces \(M_n(V)\) of \(n\) by \(n\) matrices with entries in \(V\) for every \(n= 1,2,\dots\). A matrix convex set is a collection \((K_n)_{n= 1}\) of convex sets \(K_n\subseteq M_n(V)\) satisfying the following two conditions: i) If \(n\), \(m\) are natural numbers and \(\alpha\in M_{n, m}(\mathbb{C})\) such that \(\alpha^*\alpha= 1\), then \(\alpha^*K_n\alpha\subseteq K_m\). ii) For any \(n\), \(m\) we have \(K_n\oplus K_m\subseteq K_{n+ m}\). The quantized bipolar theorem now has the following form. If \(V\) and \(V'\) are dual vector spaces, there is a natural bilinear map from \(M_n(V)\times M_n(V')\to M_{nm}(\mathbb{C})\) given, and denoted by \[ \langle\langle v, w\rangle\rangle= [\langle v_{i,j}, w_{k,\ell}\rangle]. \] The matrix polar \(K^\pi\) of a matrix convex set \(K\) is now defined by \[ K^\pi_n= \{w\in …
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46A22
Mathematics Subject Classification ID: 46A20
zbMATH DE Number: 987559
zbMATH Keywords: quantized functional analysis
zbMATH Keywords: dual vector spaces
zbMATH Keywords: bipolar theorem
zbMATH Keywords: algebraic form of the Hahn-Banach theorem
zbMATH Keywords: matrix convex set
zbMATH Keywords: quantized bipolar theorem
zbMATH Keywords: matrix polar
zbMATH Keywords: quantized form of the Hahn-Banach theorem
zbMATH Keywords: Hahn-Banach type extension theorems for completely bounded maps
reviewed by: Alfons Van Daele
MaRDI profile type: Publication
full work available at URL: jfan.1996.2958
OpenAlex ID: W2088104180
DOI: 10.1006/JFAN.1996.2958
description: scientific article; zbMATH DE number 987559