Relatively additive states on quantum logics
Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 2, pp. 327-338
Voir la notice de l'article provenant de la source Czech Digital Mathematics Library
In this paper we carry on the investigation of partially additive states on quantum logics (see [2], [5], [7], [8], [11], [12], [15], [18], etc.). We study a variant of weak states — the states which are additive with respect to a given Boolean subalgebra. In the first result we show that there are many quantum logics which do not possess any 2-additive central states (any logic possesses an abundance of 1-additive central state — see [12]). In the second result we construct a finite 3-homogeneous quantum logic which does not possess any two-valued 1-additive state with respect to a given Boolean subalgebra. This result strengthens Theorem 2 of [5] and presents a rather advanced example in the orthomodular combinatorics (see also [9], [13], [4], [6], [16], etc.). In the rest we show that Greechie logics allow for $2$-additive three-valued states, and in case of Greechie lattices we show that one can even construct many $2$-additive two-valued states. Some open questions are posed, too.
In this paper we carry on the investigation of partially additive states on quantum logics (see [2], [5], [7], [8], [11], [12], [15], [18], etc.). We study a variant of weak states — the states which are additive with respect to a given Boolean subalgebra. In the first result we show that there are many quantum logics which do not possess any 2-additive central states (any logic possesses an abundance of 1-additive central state — see [12]). In the second result we construct a finite 3-homogeneous quantum logic which does not possess any two-valued 1-additive state with respect to a given Boolean subalgebra. This result strengthens Theorem 2 of [5] and presents a rather advanced example in the orthomodular combinatorics (see also [9], [13], [4], [6], [16], etc.). In the rest we show that Greechie logics allow for $2$-additive three-valued states, and in case of Greechie lattices we show that one can even construct many $2$-additive two-valued states. Some open questions are posed, too.
Classification :
03G12, 46C05, 81P10
Keywords: (weak) state on quantum logic; Greechie paste job; Boolean algebra
Keywords: (weak) state on quantum logic; Greechie paste job; Boolean algebra
Pták, Pavel; Weber, Hans. Relatively additive states on quantum logics. Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 2, pp. 327-338. http://geodesic.mathdoc.fr/item/CMUC_2005_46_2_a7/
@article{CMUC_2005_46_2_a7,
author = {Pt\'ak, Pavel and Weber, Hans},
title = {Relatively additive states on quantum logics},
journal = {Commentationes Mathematicae Universitatis Carolinae},
pages = {327--338},
year = {2005},
volume = {46},
number = {2},
mrnumber = {2176895},
zbl = {1121.03085},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMUC_2005_46_2_a7/}
}