Measures on the quantum logic of subspaces of a \(J\)-space. (English. Russian original) Zbl 0755.46031

Sib. Math. J. 32, No. 2, 265-272 (1991); translation from Sib. Mat. Zh. 32, No. 2(186), 104-112 (1991).
The paper is devoted to a generalization of Gleason’s theorem to spaces with indefinite inner product. Let \(H\) be a space with indefinite inner product \([.,.]\) and the canonical decomposition \(H=H^ +[+]H^ -\) and the canonical symmetry \(J\) (i.e. a Krejn space). \(H\) is a Hilbert space with respect to the inner product \((x,y)=[Jx,y]\). Let \({\mathfrak B}\) be the quantum logic of \(J\)-selfadjoint bounded projections \(p\) in \(H\) and \({\mathfrak B}^ +\) (resp. \({\mathfrak B}^ -\)) all projections \(p\) with \(pH\) positive (resp. negative). Any \(p\in{\mathfrak B}\) can be decomposed as \(p=p^ ++p^ -\) with \(p^ \pm\in{\mathfrak B}^ \pm\). A map \(\mu:{\mathfrak B}\to\mathbb{R}\) is called a measure if \(\mu(p)=\sum_ i \mu(p_ i)\) for any partition \(p=\sum_ i p_ i\). The measure \(\mu\) is indefinite if \(\mu(p)\geq 0\) for \(p\in{\mathfrak B}^ +\) and \(\mu(p)\leq 0\) for \(p\in{\mathfrak B}^ -\). The main result of the paper is
Theorem. Let \(H\) be a \(J\)-space, \(\dim H\geq 3\). Then given any indefinite measure \(\mu:{\mathfrak B}\to\mathbb{R}\) there exists a unique \(J\)- selfadjoint operator \(A\) and a real \(c\in\mathbb{R}\) such that \(\mu(p)=Sp(Ap)+c\dim(p^ + H)\), \(p\in{\mathfrak B}\). Moreover if \(\dim H^ +=+\infty\) then \(c=0\) and \(0(+\infty)=0\).


46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
46C20 Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.)
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