Interval and scale effect algebras.(English)Zbl 0883.03048

A quadruple $$(E, \oplus, 0,u)$$ consisting of a set $$E$$, a partially defined binary operation $$\oplus$$, and two elements $$0,u\in E$$ is called an effect algebra if $$\oplus$$ is commutative and associative, satisfies the zero-one law (i.e. if $$p\oplus u$$ is defined, then $$p= 0)$$ and the orthosupplement law $$(\forall p\in E$$, $$\exists!q \in E$$ such that $$u= p\oplus q)$$. Given a partially ordered abelian group $$G$$ and $$u\in G$$, the interval $$G^+ [0,u]: =\{p\in G: 0\leq p \leq u\}$$ is an effect algebra with $$p\oplus q= p+q$$, defined whenever $$p+q\leq u$$. Such effect algebras are called interval algebras. In this paper it is shown that for every interval algebra $$(E, \oplus, 0,u)$$ there exists a partially ordered abelian group $$G$$ containing $$E$$ such that: $$E=G^+ [0,u]$$, every element of $$G$$ is a difference of finite sums of elements of $$E$$, and every group-valued measure $$\varphi: E\to H$$ can be extended to a group homomorphism $$\varphi^*: G\to H$$. Such a group $$G$$ is called universal ambient group for $$E$$. It is also shown that if an effect algebra admits an order-determining set of probability measures, then it is an interval algebra. Finally, scale algebras (i.e. totally ordered effect algebras) are considered. It is shown that a scale algebra $$E$$ admits a unique probability measure $$\omega$$; the latter is positive (i.e. $$\omega (p)=0$$ implies $$p=0)$$ iff $$E$$ is archimedean. Examples of nonarchimedean scale algebras are also given.

MSC:

 03G12 Quantum logic 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
Full Text:

References:

 [1] Beltrametti, E.; Cassinelli, G., The logic of quantum mechanics, Encyclopaedia of Mathematics and its Applications (1981), Addison-Wesley: Addison-Wesley Reading · Zbl 0504.03026 [2] Birkhoff, G., Lattice ordered groups, Ann. Math., 43, 298-331 (1942) · Zbl 0060.05808 [3] Busch, P.; Lahti, P.; Mittelstaedt, P., The Quantum Theory of Measurement. The Quantum Theory of Measurement, Lecture Notes in Physics, New Series M2 (1991), Springer-Verlag: Springer-Verlag Berlin [4] Darnel, M. R., Theory of Lattice-Ordered Groups (1995), Dekker: Dekker New York · Zbl 0810.06016 [5] Foulis, D. J.; Randall, C. H., What are quantum logics and what ought they to be?, (Beltrametti, E.; van Fraassen, B., Current Issues in Quantum Logic. Current Issues in Quantum Logic, Ettore Majorana International Science Series, 8 (1981), Plenum: Plenum New York), 35-52 [6] Foulis, D. J.; Bennett, M. K., Effect algebras and unsharp quantum logics, Found. Phys., 24, 1331-1352 (1994) · Zbl 1213.06004 [7] Foulis, D. J.; Greechie, R. J.; Bennett, M. K., Sums and products of interval algebras, Internat. J. Theoret. Phys., 33, 2119-2136 (1994) · Zbl 0815.06015 [8] Fuchs, L., Partially Ordered Algebraic Systems. Partially Ordered Algebraic Systems, International Series of Monographs on Pure and Applied Mathematics, 28 (1963), Pergamon: Pergamon Oxford · Zbl 0137.02001 [9] Giuntini, R.; Greuling, H., Toward a formal language for unsharp properties, Found. Phys., 19, 931-945 (1989) [10] K. Goodearl, Partially Ordered Abelian Groups with Interpolation, Mathematical Surveys and Monographs, 20, Am. Math. Soc. Providence, RI; K. Goodearl, Partially Ordered Abelian Groups with Interpolation, Mathematical Surveys and Monographs, 20, Am. Math. Soc. Providence, RI · Zbl 0589.06008 [11] Goodearl, K.; Handleman, D., Rank functions and $$K_0$$, J. Pure Appl. Algebra, 7, 195-216 (1976) · Zbl 0321.16009 [12] Gudder, S., Quantum Probability (1988), Academic Press: Academic Press Boston · Zbl 0652.60004 [13] Hamhalter, J.; Navara, M.; Pták, P., States on orthoalgebras, Internat. J. Theoret. Phys., 34, 1439-1465 (1995) · Zbl 0841.03034 [14] Kôpka, F.; Chovanec, F., $$D$$, Math. Slovaca, 44, 21-34 (1994) · Zbl 0789.03048 [15] Ludwig, G., Foundation of Quantum Mechanics (1983/1985), Springer-Verlag: Springer-Verlag New York · Zbl 0509.46057 [16] Ludwig, G., An Axiomatic Basis for Quantum Mechanics (1986/1987), Springer-Verlag: Springer-Verlag New York · Zbl 0636.46065 [17] Pták, P.; Pulmannová, S., (van der Merwe, A., Orthomodular Structures as Quantum Logics. Orthomodular Structures as Quantum Logics, Fundamental Theories of Physics, 44 (1991), Kluwer Academic: Kluwer Academic Dordrecht) · Zbl 0743.03039 [18] Piron, C., (Wightman, A., Foundations of Quantum Physics. Foundations of Quantum Physics, Mathematical Physics Monograph Series (1976), Benjamin: Benjamin Reading) · Zbl 0333.46050 [19] Prugovecki, E., Stochastic Quantum Mechanics and Quantum Space Time (1986), Reidel: Reidel Dordrecht [20] Schroeck, F.; Foulis, D., Stochastic quantum mechanics viewed from the language of manuals, Found. Phys., 20, 823-858 (1990) [21] Wilce, A., Partial abelian semigroups, Internat. J. Theoret. Phys., 34, 1807-1812 (1955) · Zbl 0839.03047
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.