Defining metric spaces via operators from unital \(C^*\)-algebras
type: Item
instance of: scholarly article
title: Defining metric spaces via operators from unital \(C^*\)-algebras
zbMATH Open document ID: 0943.46036
author: B. Pavlović
published in: Pacific Journal of Mathematics
publication date: 2000-08-23T00:00:00Z
review text: Let \(A\) be a unital \(C^{\star}\)-algebra and \(T\) an operator with \(\text{ Dom}(T)\subset A\), \(\text{ Range}(T)\) in a normed space and \(\text{Ker}(T)= {\mathbb{C}}1.\) The author considers the metric \(d_T\) on the set \({\mathcal S}(A)\) of states of \(A\) which is given by \(d_{T}(\phi , \psi) = \sup \{ |\varphi (a)-\psi (a)|: a\in A \text{ and } \|T(a) \|\leq 1\}\) for \(\varphi , \psi \in {\mathcal S}(A).\) This generalizes the definition of the metrics on \({\mathcal S}(A)\) introduced via unbounded Fredholm modules by A. Connes. The main results in this paper are two different characterizations of those operators for which the topology of \(d_T\) agrees with the week-\(\star \) topology of \({\mathcal S}(A).\)
Mathematics Subject Classification ID: 46L30
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 1344258
zbMATH Keywords: \(C^{\star }\)-algebra
zbMATH Keywords: state
zbMATH Keywords: unbounded Fredholm modules
reviewed by: José Antonio Cuenca Mira
MaRDI profile type: Publication
full work available at URL: pjm.1998.186.285
OpenAlex ID: W2169397299
DOI: 10.2140/PJM.1998.186.285
description: scientific article; zbMATH DE number 1344258