Positive extensions on \(C^*\)-algebras
type: Item
instance of: scholarly article
zbMATH Open document ID: 0985.47004
author: Ching-I Hsin
publication date: 2002-05-21T00:00:00Z
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 1599490
zbMATH Keywords: injective \(C^*\)-algebra
zbMATH Keywords: \(n\)-positive map
zbMATH Keywords: completely positive map
zbMATH Keywords: positive extensions
zbMATH Keywords: unital \(C^*\)-algebra
zbMATH Keywords: extremely disconnected topological space
MaRDI profile type: Publication
title: Positive extensions on \(C^*\)-algebras
published in: Bulletin of the Institute of Mathematics. Academia Sinica
review text: For \(k\geq 2\), let \(M_k\) be the \(C^*\)-algebra of all complex \(k\times k\) matrices. In this paper, we consider two problems related to positive extensions. One is to prove that for each unital selfadjoint subspace \({\mathcal S}\subset M_k\), and for each unital \(C^*\)-algebra \({\mathcal A}\), every \(n\)-positive map from \({\mathcal S}\) to \({\mathcal A}\) has an \(n\)-positive extension on \(M_k\) if and only if \(k=2\) and \(n=1\). The other problem is concerned about injective \(C^*\)-algebras. We prove that for unital Abelian \(C^*\)-algebras \({\mathcal B}_i\), \(1\leq i\leq k\), the \(C^*\)-algebra \(\bigoplus^k_{i=1}{\mathcal B}_i\otimes M_{n_i}\) is injective if and only if each \({\mathcal B}_i\) is isometrical \(*\)-isomorphic to \(C({\mathbf X}_i)\), where each \({\mathbf X}_i\) is a c compact Hausdorff and extremely disconnected topological space.
description: scientific article; zbMATH DE number 1599490