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Compressible groups with general comparability. (English) Zbl 1114.06012
Summary: Compressible groups generalize the order-unit space of self-adjoint operators on Hilbert space, the directed additive group of self-adjoint elements of a unital \(C^*\)-algebra, and interpolation groups with order units. In a compressible group with general comparability, each element \(g\) may be written canonically as a difference \(g = g^+ - g^-\) of elements in the positive cone \(G^+\), and the absolute value \(| g| \) is defined by \(| g| :=g^+ + g^-\). In such a group \(G\), we define and study a “pseudo-meet” \(g\mathbin {\sqcap } h\) and a “pseudo-join” \(g\mathbin {\sqcup } h\). If \(G\) is lattice-ordered, \(g\mathbin {\sqcap } h\) and \(g\mathbin {\sqcup } h\) coincide with the usual meet and join; in the general case, they retain a number of properties of the latter. We also introduce and study a so-called Rickart projection property suggested by an analogous property in Rickart \(C^*\)-algebras.

MSC:
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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References:
[1] BENNETT M. K.-FOULIS D. J.: Interval and scale effect algebras. Adv. In Appl. Math. 19 (1997), 200-215. · Zbl 0883.03048
[2] CHANG C. C. : Algebraic analysis of many-valued logics. Trans. Amer. Math. Soc 88 (1957), 467-490. · Zbl 0084.00704
[3] FOULIS D. J.: Compressible groups. Math. Slovaca 53 (2003), 433-455. · Zbl 1114.06012
[4] FOULIS D. J.: Compressions on partially ordered abelian groups. Proc Amer. Math. Soc. 132 (2000), 3581-3587. · Zbl 1063.47003
[5] GREECHIE R. J.-FOULIS D. J.-PULMANNOVÁ S.: The center of an effect algebra. Order 12 (1995), 91-106. · Zbl 0846.03031
[6] GOODEARL K. R.: Partially Ordered Abelian Groups with Interpolation. Math. Surveus Monogr. 20, Amer. Math. Soc, Providence, RI, 1986. · Zbl 0589.06008
[7] GUDDER S. P.: Examples, problems, and results in effect algebras. Internat. J. Theoret. Phus. 35 (1996), 2365-2376. · Zbl 0868.03028
[8] HARDING J.: Regularity in quantum logic. Internat. J. Theoret. Phus. 37 (1998), 1173-1212. · Zbl 0946.03077
[9] MUNDICI D.: Interpretation of \(AF\;C^ast\) -algebras in Łukasiewicz sentential calculus. J. Funct. Anal. 65 (1986), 15-63. · Zbl 0597.46059
[10] PTÁK P.-PULMANNOVÁ S.: Orthomodular Structures as Quantum Logics. Kluwer Acad. Publ., Dordrecht-Boston-London, 1991. · Zbl 0743.03039
[11] RIESZ F.-SZ.-NAGY B.: Functional Analysis. Frederick Ungar Publishing Co., New York, 1955.
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