Quantum logics with lattice state spaces
Authors:
Jiří Binder and Mirko Navara
Journal:
Proc. Amer. Math. Soc. 100 (1987), 688693
MSC:
Primary 81B10; Secondary 03G12, 06C15, 46L60
MathSciNet review:
894439
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Abstract: Let be a quantum logic and let denote the set of all states on . (By a state we mean a nonnegative bounded additive measure, not necessarily normalized.) We ask whether every logic whose state space is a lattice has to be Boolean. We prove that this is so for finite logics and "projection logics." On the other hand, we show that there exist even concrete nonBoolean logics with a lattice state space (in fact, we prove that every countable concrete logic can be enlarged to a logic with a lattice state space). In the appendix we shortly consider the lattice properties of the set of observables and correct the paper [10].
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 E. Christensen, Measures on projections and physical states, Comm. Math. Phys. 86 (1982), 529538. MR 679201 (85b:46072)
 [2]
 R. J. Greechie, Orthomodular lattices admitting no states, J. Combin. Theory Ser. A 10 (1971), 119132. MR 0274355 (43:120)
 [3]
 S. Gudder, Stochastic methods in quantum mechanics, NorthHolland, New York, 1979. MR 543489 (84j:81003)
 [4]
 R. V. Kadison, Order properties of bounded selfadjoint operators, Proc. Amer. Math. Soc. 2 (1951), 505510. MR 0042064 (13:47c)
 [5]
 P. Pták, Exotic logics, Colloq. Math. (to appear). MR 928651 (89b:03105)
 [6]
 P. Pták and J. D. M. Wright, On the concreteness of quantum logics, Apl. Mat. 30 (1985), 274285. MR 795987 (87b:03142)
 [7]
 R. M. Solovay, Axiomatic set theory (D. Scott, ed.), Proc. Sympos. Pure Math., vol 13, Part 1, Amer. Math. Soc., Providence, R.I., 1971, pp. 397428. MR 0290961 (45:55)
 [8]
 V. S. Varadarajan, Geometry of quantum theory, vol. I, Van Nostrand, Princeton, N.J., 1968. MR 0471674 (57:11399)
 [9]
 F. J. Yeadon, Finitely additive measures on projections in finite algebras, Bull. London Math. Soc. 16 (1984), 145150. MR 737242 (85i:46087)
 [10]
 N. Zierler, Order properties of bounded observables, Proc. Amer. Math. Soc. 14 (1963), 346351. MR 0145863 (26:3391)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198708944391
PII:
S 00029939(1987)08944391
Article copyright:
© Copyright 1987
American Mathematical Society
