Topology and the construction of extreme quasi-measures.

*(English)*Zbl 0914.28010Let \(X\) 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.

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 |