Pure quasi-states and extremal quasi-measures
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 \(X\).
scientific article; zbMATH DE number 537226