Communications in Mathematical Physics

, Volume 98, Issue 2, pp 187–202 | Cite as

Quantum measures and states on Jordan algebras

  • L. J. Bunce
  • J. D. Maitland Wright
Article

Abstract

A problem of Mackey for von Neumann algebras has been settled by the conjunction of the early work of Gleason and the recent advances of Christensen and Yeadon. We show that Mackey's conjecture holds in much greater generality. LetA be a JBW-algebra and letL be the lattice of all projections inA. A quantum measure onL is a countably additive map,m, fromL into the real numbers. Our results imply thatm always has a unique extension to a bounded linear functional onA, provided thatA has no TypeI2 direct summand.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aarnes, J.F.: Quasi-states onC*-algebras. Trans. Am. Math. Soc.149, 601–625 (1970)Google Scholar
  2. 2.
    Ajupov, S.A.: Extension of traces and type criterions for Jordan algebras of self-adjoint operators. Math. Z.181, 253–268 (1982)Google Scholar
  3. 3.
    Alfsen, E.M., Shultz, F.W.: State spaces of Jordan algebras. Acta. Math.140, 155–190 (1978)Google Scholar
  4. 4.
    Alfsen, E.M., Shultz, F.W., Hanché-Olsen, H.: State spaces ofC*-algebras. Acta. Math.144, 267–305 (1980)Google Scholar
  5. 5.
    Alfsen, E.M., Shultz, F.W., Størmer, E.: A Gelfand Neumark theorem for Jordan algebras. Adv. Math.28, 11–56 (1978)Google Scholar
  6. 6.
    Bunce, L.J.: Type I JB-algebras. Quart. J. Math. Oxford (2)34, 7–19 (1983)Google Scholar
  7. 7.
    Christensen, E.: Measures on projections and physical states. Commun. Math. Phys.86, 529–538 (1982)Google Scholar
  8. 8.
    Edwards, C.M.: On the facial structure of a JB-algebra. J. Lond. Math. Soc.19, 335–344 (1979)Google Scholar
  9. 9.
    Edwards, C.M.: On the centres of hereditary JBW-subalgebras of a JBW-algebra. Math. Proc. Cambs. Phil. Soc.85, 317–324 (1979)Google Scholar
  10. 10.
    Effros, E., Størmer, E.: Jordan algebras of self-adjoint operators. Trans. Am. Math. Soc.127, 312–315 (1967)Google Scholar
  11. 11.
    Gil de Lamadrid, J.: Measures and tensors. Trans. Am. Math. Soc.114, 98–121 (1965)Google Scholar
  12. 12.
    Gleason, A.M.: Measures on closed subspaces of a Hilbert space. J. Math. Mech.6, 885–893 (1957)Google Scholar
  13. 13.
    Grothendieck, A.: Produit tensoriels topologiques et espaces nucléaires. Mem. Am. Math. Soc.16 (1955)Google Scholar
  14. 14.
    Gunson, J.: Physical states on quantum logics. I. Ann. Inst. Henri Poincaré. Sect. A,XVII, 295–311 (1972)Google Scholar
  15. 15.
    Jacobson, N.: Structure and representation of Jordan algebras. Am. Math. Soc. Colloq. Publ.39. Providence, RI: Am. Math. Soc. 1968Google Scholar
  16. 16.
    Janssen, G.: Die Struktur endlicher schwach abgeschlossener Jordan algebras. II. Manuscripta Math.16, 307–332 (1975)Google Scholar
  17. 17.
    Jordan, P., von Neumann, J., Wigner, E.: On an algebraic generalisation of the quantum mechanical formalism. Ann. Math.35, 29–64 (1935)Google Scholar
  18. 18.
    Mackey, G.W.: Quantum mechanics and Hilbert space. Am. Math. Monthly64, 45–57 (1957)Google Scholar
  19. 19.
    Mackey, G.W.: The mathematical foundations of quantum mechanics. New York: Benjamin 1963Google Scholar
  20. 20.
    Matveichuk, M.S.: Description of the finite measures in semifinite algebras. Funct. Anal. Appl.15, 187–197 (1982)Google Scholar
  21. 21.
    Pedersen, G.K.:C*-algebras and their automorphism groups. New York: Academic Press 1979Google Scholar
  22. 22.
    von Neumann, J.: On an algebraic generalisation of the quantum mechanical formalism. Math. Sb.1, 415–487 (1936)Google Scholar
  23. 23.
    Sakai, S.:C*-algebras andW*-algebras. Berlin, Heidelberg, New York: Springer 1971Google Scholar
  24. 24.
    Semadeni, Z.: Barach spaces of continuous functions, Vol. I. Monog. Matematyczne, 55. Warsaw: Polish Scientific Publ. 1971Google Scholar
  25. 25.
    Shultz, F.W.: On normed Jordan algebras which are Banach dual spaces. J. Funct. Anal.31, 360–376 (1979)Google Scholar
  26. 26.
    Stacey, P.J.: TypeI 2 JBW-algebras. Quart. J. Math. Oxford (2),33 115–127 (1982)Google Scholar
  27. 27.
    Størmer, E.: On the Jordan structure ofC*-algebras. Trans. Am. Math. Soc.120, 438–447 (1965)Google Scholar
  28. 28.
    Størmer, E.: Jordan algebras of TypeI. Acta Math.115, 165–184 (1965)Google Scholar
  29. 29.
    Størmer, E.: Irreducible Jordan algebras of self-adjoint operators. Trans. Am. Math. Soc.130, 153–166 (1968)Google Scholar
  30. 30.
    Takesaki, M.: Theory of operator algebras. I. Berlin, Heidelberg, New York: Springer 1979Google Scholar
  31. 31.
    Topping, D.: Jordan algebras of self-adjoint operators. Mem. Am. Math. Soc.53 (1965)Google Scholar
  32. 32.
    Wright, J.D.M.: JordanC*-algebras. Mich. Math. J.24, 291–302 (1977)Google Scholar
  33. 33.
    Yeadon, F.J.: Measures on projections inW*-algebras of TypeII. Bull. Lond. Math. Soc.15, 139–145 (1983)Google Scholar
  34. 34.
    Yeadon, F.J.: Finitely additive measures on projections in finiteW*-algebras (to appear)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • L. J. Bunce
    • 1
  • J. D. Maitland Wright
    • 1
  1. 1.Department of MathematicsReading UniversityReadingEngland

Personalised recommendations