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On realization of generalized effect algebras. (English) Zbl 1296.03038
It is shown that a generalized effect algebra is representable as an operator in the generalized effect algebra of effects of a complex Hilbert space if and only if it has an order-determining set of generalized states.

##### MSC:
 03G12 Quantum logic 06D35 MV-algebras 06F25 Ordered rings, algebras, modules 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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##### References:
 [1] Blank, J.; Exner, P.; Havlíček, M., Hilbert space operators in quantum physics, (2008), Springer · Zbl 1163.47060 [2] H. X. Cao, Z. H. Guo, Z. L. Chen and K. L. Zhang: Representation Theory of Effect Algebras (2012), preprint, [3] Dvurečenskij, A.; Pulmannová, S., New trends in quantum structures, (2000), Kluwer Dodrecht · Zbl 0987.81005 [4] Engelking, R., General topology, revised and completed edition, (1989), Heldermann Verlag Berlin [5] Foulis, D.J.; Bennett, M.K., Effect algebras and unsharp quantum logics, Found. phys., 24, 1331-1352, (1994) · Zbl 1213.06004 [6] Greechie, R.J., Another nonstandard quantum logic (and how I found it), (), 71-85 [7] Gudder, S., Effect algebras are not adequate models for quantum mechanics, Found. phys., 40, 1566-1577, (2010) · Zbl 1218.81010 [8] J. Niederle and J. Paseka: On realization of generalized effect algebras (2012), preprint. · Zbl 1274.81010 [9] Paseka, J., PT-symmetry in (generalized) effect algebras, Internat. J. theoret. phys., 50, 1198-1205, (2011) · Zbl 1257.03094 [10] Paseka, J.; Riečanová, Z., Considerable sets of linear operators in Hilbert spaces as operator generalized effect algebras, Found. physics, 41, 1634-1647, (2011) · Zbl 1238.81009 [11] J. Paseka and Z. Riečanová: Inherited properties of effect algebras preserved by isomorphisms, preprint, 2012. [12] Polakovič, M., Generalized effect algebras of bounded positive operators defined on Hilbert spaces, Rep. math. phys., 68, 241-250, (2011) · Zbl 1250.81014 [13] Polakovič, M.; Riečanová, Z., Generalized effect algebras of positive operators densely defined on Hilbert space, Internat. J. theor. phys., 50, 1167-1174, (2011) · Zbl 1237.81009 [14] Riečanová, Z.; Zajac, M.; Pulmannová, S., Effect algebras of positive operators densely defined on Hilbert space, Rep. math. phys., 68, 261-270, (2011) · Zbl 1250.81015 [15] Riečanová, Z.; Zajac, M., Hilbert space effect-representations of effect algebras, Rep. math. phys., 70, 283-290, (2012) · Zbl 1268.81014
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