zbMATH — the first resource for mathematics

Convex and linear effect algebras. (English) Zbl 0956.46002
In the paper it is shown that convex effect algebras arise naturally in the description of physical statistical system. It is proved that a convex effect algebra \(P\) posseses a separating (resp. an order determining) set of states if and only if the corrsponding ordered linear space \((V,K)\) has no nonzero \(u\)-infinitesimals (resp. \((V,K)\) is an order unit space). Using these assertions it is shown that an effect algebra \(P\) is imbeddable in an interval \([\theta,u]\) of an order unit space if and only if the state space of \(P\) is order determining. An element \(a\) of effect algebra is sharp (resp. extremal) if \(a\wedge a'=0\) (resp. \(a=\lambda b\oplus (1-\lambda)c\) for \(\lambda\in (0,1)\) implies that \(b=c\)). Some properties of sharp and extremal elements are considered. One of the basic result is the following Theorem.
Suppose that the state space of \([\theta,u]\) is sharply determining.
(i) Every sharp element of \([\theta,u]\) is extremal and hence the sharp and extreme elements of \([\theta,u]\) coincide.
(ii) If \(a,b\in [\theta,u]\) are sharp and \(a\perp b\) then \(a\oplus b\) is sharp.
An alternative definition of a convex effect algebra called a \(CE\)-algebra is considered. It is shown that a convex effect algebra \(P\) is an \(MV\)-algebra if and only if \(P\) is lattice ordered.

46A40 Ordered topological linear spaces, vector lattices
46B40 Ordered normed spaces
46N50 Applications of functional analysis in quantum physics
Full Text: DOI
[1] Beltrametti, E.G; Bugajski, S, J. phys. A: math. gen., 28, 3329-3343, (1995) · Zbl 0859.46049
[2] Beltrametti, E.G; Bugajski, S, Int. J. theor. phys., 34, 1221-1229, (1995)
[3] Beltrametti, E.G; Bugajski, S, J. math. phys., 38, 3020-3030, (1997)
[4] Bugajski, S, Int. J. theor. phys., 35, 2229-2244, (1996)
[5] Bugajski, S; Hellwig, K.-E; Stulpe, W, Rep. math. phys., 41, 1-11, (1998)
[6] Busch, P; Grabowski, M; Lahti, P, Operational quantum physics, (1995), Springer Berlin · Zbl 0863.60106
[7] Busch, P; Lahti, P; Mittlestaedt, P, The quantum theory of measurement, (1991), Springer Berlin
[8] Cattaneo, G; Nisticò, G, Nuovo cimenta, 90B, 161-175, (1985)
[9] Chang, C.C, Trans. amer. math. soc., 88, 467-490, (1958)
[10] Davies, E.B, Quantum theory of open systems, (1976), Academic Press London · Zbl 0388.46044
[11] Dvurečenskij, A, Trans. amer. math. soc., 147, 1043-1057, (1995)
[12] Dvurečenskij, A; Pulmannová, S, Int. J. theor. phys., 33, 819-850, (1994)
[13] Edwards, R.E, Functional analysis, (1965), Holt, Rinehart and Winston New York · Zbl 0182.16101
[14] Foulis, D; Bennett, M.K, Found. phys., 24, 1331-1352, (1994)
[15] Foulis, D; Bennett, M.K, Advances appl. math., 19, 200-215, (1997)
[16] Giuntini, R; Greuling, H, Found. phys., 19, 931-945, (1989)
[17] Greechie, R; Foulis, D, Int. J. theor. phys., 34, 1369-1382, (1995)
[18] Gudder, S, Found. phys., 26, 813-822, (1996)
[19] Gudder, S, Int. J. theor. phys., 35, 2365-2376, (1996)
[20] Gudder, S, Demon. math., 31, 235-254, (1998)
[21] S. Gudder: Convex structures and effect algebras (to appear). · Zbl 0963.03085
[22] S. Gudder and S. Pulmannová: Representation theorem for convex effect algebras, Comment. Math. Univ. Carolinae (to appear).
[23] Holevo, A.S, Probabilistic and statistical aspects of quantum theory, (1982), North-Holland Amsterdam · Zbl 0497.46053
[24] Kôpka, F, Tatra mountains math. publ., 1, 83-87, (1992)
[25] Kôpka, F; Chovanec, F, Math. slovaca, 44, 21-34, (1994)
[26] Kôpka, F; Chovanec, F, Tatra mountains math. publ., 10, 183-197, (1997)
[27] Kraus, K, States, effects, and operations, (1983), Springer-Verlag Berlin
[28] Ludwig, G, Foundations of quantum mechanics, (1983), Springer-Verlag Berlin · Zbl 0509.46057
[29] Miles, P, Trans. amer. math. soc., 107, 217-236, (1963)
[30] Mundici, D, J. funct. anal., 65, 15-63, (1986)
[31] Namioka, I, Partially ordered linear topological spaces, memoirs, (1957), Amer. Math. Soc 24, Providence, Rhode Island · Zbl 0105.08901
[32] Pulmannová, S, Int. J. theor. phys., 34, 189-207, (1995)
[33] Pulmannová, S, Demon. math., 30, 313-328, (1997)
[34] Rudolf, O, J. math. phys., 37, 5368-5379, (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.