# zbMATH — the first resource for mathematics

Effect algebras of positive linear operators densely defined on Hilbert spaces. (English) Zbl 1250.81015
Summary: We show that the set of all positive linear operators densely defined in an infinite-dimensional complex Hilbert space can be equipped with a partial sum of operators making it a generalized effect algebra. This sum coincides with the usual sum of two operators whenever it exists. Moreover, blocks of this generalized effect algebra are proper sub-generalized effect algebras. All intervals in this generalized effect algebra become effect algebras which are Archimedean, convex, interval effect algebras, for which the set of vector states is order determining. Further, these interval operator effect algebras possess faithful states.

##### MSC:
 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 47B48 Linear operators on Banach algebras 81P16 Quantum state spaces, operational and probabilistic concepts
Full Text:
##### References:
 [1] Blank, J.; Exner, P.; Havliček, M., Hubert space operators in quantum physics, (2008), Springer [2] Bugajski, S.; Gudder, S.; Pulmannová, S., Convex effect algebras, state ordered effect algebras and ordered linear spaces, Rep. math. phys., 45, 371-388, (2000) · Zbl 0963.46004 [3] Gudder, S.; Pulmannová, S., Representation theorem for convex effect algebras, Commet. math. univ. carolinae, 39, 645-659, (1998) · Zbl 1060.81504 [4] Gudder, S.; Pulmannová, S.; Bugajski, S.; Beltrametti, E., Convex and linear effect algebras, Rep. math. phys., 44, 359-379, (1999) · Zbl 0956.46002 [5] Dvurečenskij, A.; Pulmannová, S., (), Mathematics and its Applications (Dordrecht) and Ister Science, Bratislava [6] Foulis, D.J.; Bennet, M.K., Effect algebras and unsharp quantum logics, Found. phys., 24, 1331-1352, (1994) · Zbl 1213.06004 [7] Hedlíková, J.; Pulmannová, S., Generalized difference posets and orthoalgebras, Acta math. univ. comenianae, LXV, 247-279, (1996) · Zbl 0922.06002 [8] Kalmbach, G.; Riečanová, Z., An axiomatization for abelian relative inverses, Demonstratio math., 27, 769-780, (1996) · Zbl 0826.08002 [9] Kôpka, F.; Chovanec, F., D-posets, Math. slovaca, 44, 21-34, (1994) · Zbl 0789.03048 [10] Polakovič, M.; Riečanová, Z., Generalized effect algebras of positive operators densely defined on hubert space, Int. J. theor phys., 50, 1167-1174, (2011) · Zbl 1237.81009 [11] Riečanová, Z., Proper effect algebras admitting no states, Internat. J. theor. phys., 40, 1683-1691, (2001) · Zbl 0989.81003
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.