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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 $$X$$.

##### MSC:
 46J25 Representations of commutative topological algebras 46J99 Commutative Banach algebras and commutative topological algebras 46E15 Banach spaces of continuous, differentiable or analytic functions
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##### References:
 [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
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