zbMATH — the first resource for mathematics

Pure quasi-states and extremal quasi-measures. (English) Zbl 0791.46028
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 \(X\).

46J25 Representations of commutative topological algebras
46J99 Commutative Banach algebras and commutative topological algebras
46E15 Banach spaces of continuous, differentiable or analytic functions
Full Text: DOI EuDML
[1] Aarnes, J.F.: Quasi-states and quasi-measures. Adv. Math.86, 41-67 (1991) · Zbl 0744.46052
[2] Aarnes, J.F.: The construction of general quasi-measures in locally compact spaces (to appear)
[3] Aarnes, J.F.: Quasi-states on C*-algebras. Trans. Am. Math. Soc.149, 601-625 (1970) · Zbl 0212.15403
[4] Choquet, J.: Lectures on analysis, vol. 1. New York: Benjamin 1969 · Zbl 0181.39602
[5] Dixmier, J.: Les C*-algebres et leurs representations. Paris: Gauthier-Villars 1964 · Zbl 0152.32902
[6] Knudsen, F.F.: Topology and the construction of extreme Quasi-measures (to appear) · Zbl 0914.28010
[7] Phelps, R.R.: Lectures on Choquet’s theorem. (Math. Stud., vol. 7) Princeton: van Nostrand 1966 · Zbl 0135.36203
[8] Rudin, W.: Functional analysis. New York: McGraw-Hill 1973 · Zbl 0253.46001
[9] Rudin, W.: Real and complex analysis, third ed. New York: McGraw-Hill 1987 · Zbl 0925.00005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.