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On the Lebesgue decomposition of Gleason measures. (English) Zbl 0621.46058
The following theorem is proved: Let \(\omega\) and m be finite signed measures on the lattice L of all closed subspaces of a separable Hilbert space and let \(\omega\) be nonnegative. Then there are signed measures \(m_ 1\), \(m_ 2\) such that \(m=m_ 1+m_ 2\), \(m_ 1\) is dominated by \(\omega\) (i.e., \(\omega (a)=0\Rightarrow m_ 1(a)=0)\) and \(m_ 2\) is singular to \(\omega\) (i.e., there is an element \(a_ 0\in L\) such that \(a\leq a_ 0\Rightarrow \omega (a)=0\) and \(b\leq a_ 0^{\perp}\Rightarrow m_ 2(b)=0)\). In contrast to the classical Lebesgue decomposition, \(m_ 1\), \(m_ 2\) cannot be chosen nonnegative if m is nonnegative.
Reviewer: M.Navara

46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
46C99 Inner product spaces and their generalizations, Hilbert spaces
06C15 Complemented lattices, orthocomplemented lattices and posets
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