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Topology and the construction of extreme quasi-measures. (English) Zbl 0914.28010
Let $$X$$ be a compact Hausdorff space. In connection with the problem when a quasi-state on $$C(X)$$ is a state, J. F. Aarnes [Adv. Math. 86, No. 1, 41-67 (1991; Zbl 0744.46052)] introduced the concept of a quasi-measure in $$X$$. That is a real-valued, nonnegative, monotone function $$\mu$$ on the system $${\mathcal A}(X)$$ of all closed and all open subsets of $$X$$ such that $$\mu(A_1\cup A_2)= \mu(A_1)+ \mu(A_2)$$ for disjoint sets $$A_1,A_2\in{\mathcal A}(X)$$ and $$\mu(A)= \sup\{\mu(F)$$: $$F\subseteq A$$, $$F$$ is closed in $$X\}$$ for open subsets $$A$$ of $$X$$. A two-valued quasi-measure is called extreme.
The main result of the paper under review says that an extreme quasi-measure on a connected locally connected, compact Hausdorff space is uniquely determined by its restriction to the family $${\mathcal T}(X)$$ of all closed, connected and co-connected subsets of $$X$$; here a set is called co-connected if its complement is connected. Moreover, the author characterizes the partitions $${\mathcal T}_0$$, $${\mathcal T}_1$$ of $${\mathcal T}(X)$$ for which $${\mathcal T}_0= \{T\in{\mathcal T}(X):\mu(T)= 0\}$$ and $${\mathcal T}_1= \{T\in{\mathcal T}(X): \mu(T)= 1\}$$ for some extreme quasi-measure $$\mu$$. Some nontrivial quasi-measures on the torus $$T^2$$ are constructed.
Reviewer: H.Weber (Udine)

MSC:
 28C15 Set functions and measures on topological spaces (regularity of measures, etc.) 46L30 States of selfadjoint operator algebras
Zbl 0744.46052
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