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The Mackey-Gleason problem


Authors: L. J. Bunce and J. D. Maitland Wright
Journal: Bull. Amer. Math. Soc. 26 (1992), 288-293
MSC (2000): Primary 46L50
DOI: https://doi.org/10.1090/S0273-0979-1992-00274-4
MathSciNet review: 1121569
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Abstract: Let A be a von Neumann algebra with no direct summand of Type $ {{\text{I}}_2}$, and let $ \mathcal{P}(A)$ be its lattice of projections. Let X be a Banach space. Let $ m:\mathcal{P}(A) \to X$ be a bounded function such that $ m(p + q) = m(p) + m(q)$ whenever p and q are orthogonal projections. The main theorem states that m has a unique extension to a bounded linear operator from A to X. In particular, each bounded complex-valued finitely additive quantum measure on $ \mathcal{P}(A)$ has a unique extension to a bounded linear functional on A.


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  • [1] J. F. Aarnes, Quasi-states on $ {C^{\ast}}$ algebras, Trans. Amer. Math. Soc. 149 (1970), 601-625. MR 0282602 (43:8311)
  • [2] L. J. Bunce and J. D. M. Wright, Complex measures on projections in von Neumann algebras, J. London Math. Soc. (2) (to appear). MR 1182483 (93j:46069)
  • [3] -, Continuity and linear extensions of quantum measures on Jordan operator algebras, Math. Scand. 64 (1989), 300-306). MR 1037464 (91f:46096)
  • [4] -, The Mackey-Gleason problem for vector measures on projections in a von Neumann algebra, submitted.
  • [5] -, Quantum logic, state space geometry and operator algebras, Comm. Math. Phys. 96 (1984), 345-348. MR 769351 (86e:81017)
  • [6] -, Quantum measures and states on Jordan algebras, Comm. Maths. Phys. 98 (1985), 187-202. MR 786572 (86k:46101)
  • [7] E. Christensen, Measures on projections and physical states, Comm. Math. Phys. 86 (1982), 529-538. MR 679201 (85b:46072)
  • [8] R. Cooke, M. Keane, and W. Moran, An elementary proof of Gleason's Theorem, Math. Proc. Cambridge Philos. Soc. 98 (1985), 117-128. MR 789726 (86h:46098)
  • [9] A. M. Gleason, Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6 (1957), 885-893. MR 0096113 (20:2609)
  • [10] J. Gunson, Physical states on quantum logics I, Ann. Inst. H. Poincaré 17 (1972), 295-311. MR 0336364 (49:1139)
  • [11] G. W. Mackey, The mathematical foundations of quantum mechanics, Benjamin, 1963.
  • [12] S. Maeda, Probability measures on projections in von Neumann algebras, Reviews in Mathematical Physics 1 (1990), 235-290. MR 1070091 (92m:46100)
  • [13] A. Paszkiewicz, Measures on projections in $ {W^{\ast}}$-factors, J. Funct. Anal. 62 (1985), 87-117. MR 790772 (86m:46060)
  • [14] M. Takesaki, Theory of operator algebras, Springer, 1979. MR 548728 (81e:46038)
  • [15] F. J. Yeadon, Finitely additive measures on projections in finite $ {W^{\ast}}$-algebras, Bull. London Math. Soc. 16 (1984), 145-150. MR 737242 (85i:46087)
  • [16] -, Measures on projections in $ {W^{\ast}}$-algebras of Type $ {\text{I}}{{\text{I}}_1}$, Bull. London Math. Soc. 15 (1983), 139-145. MR 689246 (84g:46089)

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DOI: https://doi.org/10.1090/S0273-0979-1992-00274-4
Article copyright: © Copyright 1992 American Mathematical Society

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