Probability measures in $W^{*}J$-algebras in Hilbert spaces with conjugation
HTML articles powered by AMS MathViewer
- by Marjan Matvejchuk PDF
- Proc. Amer. Math. Soc. 126 (1998), 1155-1164 Request permission
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 $\mu (p)=\tau (p)$, $\forall p\in P$.References
- Andrew M. Gleason, Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6 (1957), 885–893. MR 0096113, DOI 10.1512/iumj.1957.6.56050
- K. Yu. Dadashyan and S. S. Khoruzhiĭ, Field algebras in quantum theory with indefinite metric, Teoret. Mat. Fiz. 54 (1983), no. 1, 57–77 (Russian, with English summary). MR 704009
- M. S. Matveĭchuk, Measures on the quantum logic of subspaces of a $J$-space, Sibirsk. Mat. Zh. 32 (1991), no. 2, 104–112, 212 (Russian); English transl., Siberian Math. J. 32 (1991), no. 2, 265–272. MR 1138445, DOI 10.1007/BF00972773
- M. S. Matveĭchuk, Description of indefinite measures on $W^\ast J$-factors, Dokl. Akad. Nauk SSSR 319 (1991), no. 3, 558–561 (Russian); English transl., Soviet Math. Dokl. 44 (1992), no. 1, 161–165. MR 1148972
- Marjan Matvejchuk, Semiconstant measures on hyperbolic logics, Proc. Amer. Math. Soc. 125 (1997), no. 1, 245–250. MR 1350955, DOI 10.1090/S0002-9939-97-03591-0
- Sh. A. Ayupov, Klassifikatsiya i predstavlenie uporyadochennykh ĭ ordanovykh algebr, “Fan”, Tashkent, 1986 (Russian). MR 921432
- T. Ya. Azizov and I. S. Iokhvidov, Linear operators in spaces with an indefinite metric, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1989. Translated from the Russian by E. R. Dawson; A Wiley-Interscience Publication. MR 1033489
- M. S. Matvejchuk, Linearity of Charges on the Lattice of Projections, Russian Math. (Iz. VUZ) 39 (1995), 48–66 (Russian).
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
- 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
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 1155-1164
- MSC (1991): Primary 81P10, 46L50, 46B09, 46C20, 03G12; Secondary 28A60
- DOI: https://doi.org/10.1090/S0002-9939-98-04176-8
- MathSciNet review: 1425135