The paper is devoted to the investigation of singular states on topological *-algebras of unbounded operators defined on (F)-domains. There are considered only functionals which are continuous with respect to the so called uniform topology τD. Equivalent characterizations of positive singular functional and a decomposition result for functionals are proved. Analogously to the bounded case positive singular functional can be given with the help of limits on free ultrafilters. Moreover, the state space of the maximal Op*-algebra is the w*-closed convex hull of vector states (pure states) - despite the fact that the state space is not w-compact.
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Frank Löffler, Werner Timmermann, Singular States on Maximal Op*-algebras. Publ. Res. Inst. Math. Sci. 22 (1986), no. 4, pp. 671–687DOI 10.2977/PRIMS/1195177627