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)

 
 

 

States on $ W\sp *$-algebras and orthogonal vector measures


Author: Jan Hamhalter
Journal: Proc. Amer. Math. Soc. 110 (1990), 803-806
MSC: Primary 81P10; Secondary 46L30
DOI: https://doi.org/10.1090/S0002-9939-1990-1036987-X
MathSciNet review: 1036987
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that every state on a $ {W^ * }$-algebra $ \mathcal{A}$ without type $ {I_2}$ direct summand is induced by an orthogonal vector measure on $ \mathcal{A}$. This result may find an application in quantum stochastics $ [1,7]$. Particularly, it allows us to find a simple formula for the transition probability between two states on $ \mathcal{A}$ $ [3,8]$.


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

  • [1] A. Dvurečenskij and S. Pulmannová, Random measures on a logic, Demonstratio Math. 14 (1981), 305-320. MR 632289 (82k:81005)
  • [2] A. Gleason, Measures on closed subspaces of a Hilbert space, J. Math. Mech. 6 (1965), 428-442. MR 0096113 (20:2609)
  • [3] S. P. Gudder, Quantum probability, Academic Press, New York, 1988. MR 958911 (90c:81013)
  • [4] -, Some unsolved problems in quantum logics, Mathematical Foundations of Quantum Theory, Academic Press, New York, 1978. MR 0495813 (58:14461)
  • [5] J. Hamhalter and P. Pták, Hilbert-space-valued states on quantum logics (to appear).
  • [6] E. Christensen, Measures on projections and physical states, Commun. Math. Phys. 86 (1982), 529-538. MR 679201 (85b:46072)
  • [7] R. Jajte and A. Paszkiewicz, Vector measures on the closed subspaces of a Hilbert space, Studia Math. 63 (1978), 229-251. MR 0632053 (58:30225)
  • [8] P. Kruszyński, Vector measures on orthocomplemented lattices, Proc. of the Koninklijke Nederlandske Akademie van Wetenschappen, Ser. A (4) 91 (1988), 427-442. MR 976526 (90a:46104)
  • [9] R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras, Academic Press, New York, 1986. MR 859186 (88d:46106)
  • [10] F. J. Yeadon, Measures on projections in $ {W^ * }$-algebras of type $ II_{1}$, Bull. London Math. Soc. 15 (1983), 139-145. MR 689246 (84g:46089)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 81P10, 46L30

Retrieve articles in all journals with MSC: 81P10, 46L30


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1036987-X
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society