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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Probability measures in $W^{*}J$-algebras
in Hilbert spaces with conjugation


Author: Marjan Matvejchuk
Journal: Proc. Amer. Math. Soc. 126 (1998), 1155-1164
MSC (1991): Primary 81P10, 46L50, 46B09, 46C20, 03G12; Secondary 28A60
MathSciNet review: 1425135
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Abstract: Let $\mathcal{M}$ be a real $W^{*}$-algebra of $J$-real bounded operators containing no central summand of type $I_{2}$ in a complex Hilbert space $H$ with conjugation $J$. Denote by $P$ the quantum logic of all $J$-orthogonal projections in the von Neumann algebra ${\mathcal{N}}={\mathcal{M}}+ i{\mathcal{M}}$. Let $\mu :P\rightarrow [0,1]$ be a probability measure. It is shown that $\mathcal{N}$ contains a finite central summand and there exists a normal finite trace $\tau $ on $\mathcal{N}$ such that $\mu (p)=\tau (p)$, $\forall p\in P$.


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Additional Information

Marjan Matvejchuk
Affiliation: Department of Mechanics and Mathematics, Kazan State University, 18 Lenin St., 420008, Kazan, Russia
Address at time of publication: Department of Physics and Mathematics, Ulyanovsk Pedagogical University, 432700 Ulyanovsk, Russia
Email: Marjan.Matvejchuk@ksu.ru

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04176-8
PII: S 0002-9939(98)04176-8
Keywords: Quantum logics, measure, Hilbert space, $W^*$-algebra
Received by editor(s): April 12, 1996
Received by editor(s) in revised form: October 7, 1996
Additional Notes: The research described in this paper was made possible in part by Grant N:1 of the Russian Government “Plati Sebe Sam" and was supported by the Russian Foundation for Basic Research (grant 96-01-01265)
Communicated by: Dale Alspach
Article copyright: © Copyright 1998 American Mathematical Society