Pure quasi-states and extremal quasi-measures
type: Item
instance of: scholarly article
title: Pure quasi-states and extremal quasi-measures
zbMATH Open document ID: 0791.46028
DOI: 10.1007/BF01444904
author: Johan F. Aarnes
published in: Mathematische Annalen
publication date: 1994-03-23T00:00:00Z
full work available at URL: https://eudml.org/doc/165059
review text: Let \(X\) be a compact Hausdorff space and let \(C(X)\) denote the space of continuous real-valued functions on \(X\). A quasi-state is a function \(\rho: C(X)\to \mathbb{R}\) such that \(\rho(a)\geq 0\) if \(a\geq 0\), \(a\in C(X)\), \(\rho(1) =1\) and \(\rho\) is linear on each closed subalgebra \(A(a)\) generated by a single element \(a\in C(X)\). The set of all quasi-states \(Q\) is convex, and is compact in the topology of pointwise convergence on \(C(X)\). Associated to each quasi-state is a set-function in \(X\) which is defined only on the closed sets and the open sets in \(X\). This set function is called a quasi-measure, and as in general not sub- additive. A quasi-measure is called extremal if it only takes the values 0 and 1. The first part of the paper is concerned with characterizing the extremal quasi-measures and their associated quasi-states, which are called simple. The set \(E\) of simple quasi-states is a proper subset of the set \(Q_ e\) of extreme points in \(Q\). The crucial property, however, is that a quasi-state is simple if and only if it is multiplicative on \(A(a)\) for each \(a\in A\). This makes it possible to show that \(E\) is closed in \(Q\) so that \(E\) is a compact Hausdorff space. On the basis of this we introduce a ``nonlinear Gelfand-transform'' \(\Psi\) of \(C(X)\) into \(C(E)\) defined by \(\Psi(a)(\sigma)= \sigma(a)\); \(a\in C(X)\), \(\sigma\in E\). \(\Psi\) is an isometric, order-preserving nonlinear map. It turns out that each quasi-state \(\rho\) is the closed convex hull of \(E\) may be factored as \(\rho= p\circ \Psi\), where \(p\) is an ordinary linear state on \(C(E)\). In general, this factorization is non-unique, as shown in an example. This non-uniqueness reflects that the order-structure of the positive cone generated by \(Q\) generally is quite complicated, and is closely bound up with the connectivity-properties of the space \(X\).
Mathematics Subject Classification ID: 46J25
Mathematics Subject Classification ID: 46J99
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 537226
zbMATH Keywords: nonlinear Gelfand-transform
zbMATH Keywords: extremal quasi-measures
zbMATH Keywords: simple quasi- states
zbMATH Keywords: order-structure of the positive cone
zbMATH Keywords: connectivity-properties
MaRDI profile type: Publication
cites work: Quasi-states and quasi-measures
cites work: Quasi-States on C ∗ -Algebras
cites work: Q5571891
cites work: Q5531996
cites work: Topology and the construction of extreme quasi-measures
cites work: Q5508589
cites work: Q5667036
cites work: Q4340161
OpenAlex ID: W1991506249
description: scientific article; zbMATH DE number 537226