We analyze finitely additive orthogonal states whose values lie in a real Hilbert space. We call them $h$-states. We first consider the important case of $h$-states on a standard Hilbert logic $L(H)$ of projectors in $H$-we describe the $h$-states $s$: $L(H_1) \rightarrow H_2$, where $\text {dim } H_2 \leq$ \text {dim} H_1 < \infty$. In particular, we show that, up to a unitary mapping, every $h$-state $s$: $L(H)\rightarrow H(3\leq \text {dim } H < \infty)$ has to be concentrated on a one-dimensional projection. We also study the $h$-states $s$: $L(H_1)\rightarrow H_2$ for the case of $\text {dim } H_1 = \infty$. The results of the first part complement the papers [10] and [13]. In the second part we investigate $h$-states on general logics. Being motivated by the quantum axiomatics, the main question we ask here is as follow: Given a Hilbert space $H$ with $\text {dim } H < \infty$, what is the class of such logics $L$ that, for any Boolean subalgebra $B$ of $L$, every $h$-states $s$: $B \rightarrow H$ extends over $L$? We answer this question by finding a simple condition characterizing the class (Theorem 3.4]. It turns out that the class is considerably large-it contains e.g. all concrete logics-but, on the other hand, it does not contain all finite logics (we construct a counterexample in the appendix).
@article{104491,
author = {Jan Hamhalter and Pavel Pt\'ak},
title = {Hilbert-space-valued states on quantum logics},
journal = {Applications of Mathematics},
volume = {37},
year = {1992},
pages = {51-61},
zbl = {0767.03034},
mrnumber = {1152157},
language = {en},
url = {http://dml.mathdoc.fr/item/104491}
}
Hamhalter, Jan; Pták, Pavel. Hilbert-space-valued states on quantum logics. Applications of Mathematics, Tome 37 (1992) pp. 51-61. http://gdmltest.u-ga.fr/item/104491/
On 0-1 measure for projectors II, Aplikace matematiky 26 (1981), 57-58. (1981) | MR 0602402 | Zbl 0459.28020
Fundamentals of the theory of operators algebras, Vol. I, Academic Press, Inc., 1986. (1986) | MR 0859186
Random measures on a logic, Demonstratio Math XIV no. 2 (1981). (1981) | MR 0632289
Can quantum-mechanical description of reality be considered complete?, Phys. Rev. 47 (1935), 777-780. (1935) | Article
Measures on the closed subspaces of Hilbert space, J. Math. Mech. 65 (1957), 885-893. (1957) | MR 0096113
Orthomodular lattices admitting no states, Journ. Comb. Theory 10 (1971), 119-132. (1971) | Article | MR 0274355 | Zbl 0219.06007
Stochastic Methods in Quantum Mechanics, North Holland, New York, 1979. (1979) | MR 0543489 | Zbl 0439.46047
Dispersion-free states and the existence of hidden variables, Proc. Amer. Math. Soc 19 (1968), 319-324. (1968) | Article | MR 0224339
Measures in Boolean algebras, Trans. Amer. Math. Soc. 64 (1948), 467-497. (1948) | Article | MR 0028922
Vector measures on the closed subspaces of a Hilbert space, Studia Math. T. LXIII (1978). (1978) | MR 0632053
Measures on Hilbert Lattices, World Sci. Publ., Singapore, 1986. (1986) | MR 0867884
Vector measures on orthocomplemented lattices, Proceedings of the Koniklijke Akademie van Wetenschappen, Ser. A, 91 no. 4, December 19 (1988). (1988) | MR 0976526
Classes equationelles de trellis orthomodulaires et espaces de Hilbert, These pour obtenir Docteur d'Etat es Sciences, Université Claude Bernard-Lyon (France), 1987. (1987)
Exotic logics, Colloqium Mathematicum LIV, Fasc. 1 (1987), 1-7. (1987) | MR 0928651
Weak dispersion-free states and the hidden variables hypothesis, J. Math. Physics 24 (1983), 839-841. (1983) | Article | MR 0700618
On the concreteness of quantum logics, Aplikace matematiky 30 č. 4 (1986), 274-285. (1986) | MR 0795987
A characterization of state space of orthomodular lattices, Jour. Comb. Theory 17 (1974), 317-328. (1974) | Article | MR 0364042