Quantum logics with lattice state spaces

Authors:
Jiří Binder and Mirko Navara

Journal:
Proc. Amer. Math. Soc. **100** (1987), 688-693

MSC:
Primary 81B10; Secondary 03G12, 06C15, 46L60

DOI:
https://doi.org/10.1090/S0002-9939-1987-0894439-1

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 non-Boolean 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**].

**[1]**E. Christensen,*Measures on projections and physical states*, Comm. Math. Phys.**86**(1982), 529-538. MR**679201 (85b:46072)****[2]**R. J. Greechie,*Orthomodular lattices admitting no states*, J. Combin. Theory Ser. A**10**(1971), 119-132. MR**0274355 (43:120)****[3]**S. Gudder,*Stochastic methods in quantum mechanics*, North-Holland, New York, 1979. MR**543489 (84j:81003)****[4]**R. V. Kadison,*Order properties of bounded self-adjoint operators*, Proc. Amer. Math. Soc.**2**(1951), 505-510. 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), 274-285. 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. 397-428. 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), 145-150. MR**737242 (85i:46087)****[10]**N. Zierler,*Order properties of bounded observables*, Proc. Amer. Math. Soc.**14**(1963), 346-351. MR**0145863 (26:3391)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1987-0894439-1

Article copyright:
© Copyright 1987
American Mathematical Society