Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. A. M. Gleason, Measures on the closed subspaces of a Hilbert space, J. Math. Mech 6 (1957), 885-893. MR 20:2609
  • 2. K. Yu. Dadashyan and S.S. Horujy, On Field algebras in quantum theory with indefinite metric, Theor. and Math. Phys 54 (1983), 57-77 (Russian). MR 85d:81082
  • 3. M. S. Matvejchuk, Measure on quantum logics of subspaces of a J-space, Siberian Mathem. J. 32 (1991), 265-272 (Russian). MR 92j:46137
  • 4. M. S. Matvejchuk, A description of indefinite measures in $W^{*}J$-factors, Soviet Math. Dokl. 44 (1992), 161-165 (Russian). MR 93a:46123
  • 5. M. S. Matvejchuk, Semiconstant measures on hyperbolic logics, Proceedings of the American Mathematical Society 125 (1997), 245-250. MR 97c:46081
  • 6. Sh. A. Ayupov, Classification and representation of ordered Jordan algebras, Fan, Tashkent, Uzbekistan, UdSSR, 1986 (Russian). MR 89b:46083
  • 7. T. Ya. Azizov and I. S. Iokhvidov, Linear operators in space with an indefinite metric, Wiley, New York, 1989 (Russian). MR 90j:47042
  • 8. M. S. Matvejchuk, Linearity of Charges on the Lattice of Projections, Russian Math. (Iz. VUZ) 39 (1995), 48-66 (Russian).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 81P10, 46L50, 46B09, 46C20, 03G12, 28A60

Retrieve articles in all journals with MSC (1991): 81P10, 46L50, 46B09, 46C20, 03G12, 28A60

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

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

American Mathematical Society