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Jauch-Piron states on von Neumann algebras. (English) Zbl 0791.46043

We give a detailed analysis of Jauch-Piron states on a von Neumann algebra \(M\). When \(M\) is \(\sigma\)-finite it is shown that all Jauch-Piron states are non-singular if \(M\) is properly infinite and are regular if \(M\) is Type III whereas, if \(M\) is finite, singular Jauch-Piron states often exist. Non-\(\sigma\)-finite algebras \(M\) are also discussed and it is shown that Jauch-Piron pure states are \(\sigma\)-additive whenever \(M\) has no abelian part.

MSC:

46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
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[1] Amann, A.: Jauch-Piron states inW *-algebraic quantum mechanics. J. Math. Phys.28 (10), 2384–2389 (1989) · Zbl 0637.46074 · doi:10.1063/1.527775
[2] Bunce, L.J., Wright, J.D.M.: Complex measures on projections in von Neumann algebras. J. Lond. Math. Soc. (to appear) · Zbl 0724.46049
[3] Bunce, L.J., Wright, J.D.M.: The Mackey-Gleason problem. Bull. Am. Math. Soc.26 (no. 2), 288–293 (1992) · Zbl 0759.46054 · doi:10.1090/S0273-0979-1992-00274-4
[4] Bunce, L.J., Navara, M., Pták, P., Wright, J.D.M.: Quantum logics with Jauch-Piron states. Q. J. Math., Oxf. II. Ser.36, 261–272 (1985) · Zbl 0585.03038 · doi:10.1093/qmath/36.3.261
[5] Christensen, E.: Measures on projections and physical states. Commun. Math. Phys.86, 529–538 (1982) · Zbl 0507.46052 · doi:10.1007/BF01214888
[6] Dixmier, J.:C *-Algebras. Amsterdam: North Holland 1977 · Zbl 0372.46058
[7] Gleason, A.M.: Measures on closed subspaces of a Hilbert space. J. Math. Mech.6, 885–893 (1957) · Zbl 0078.28803
[8] Jauch, J.M.: Foundations of Quantum Mechanics. Reading, MA: Addison-Wesley 1968 · Zbl 0166.23301
[9] Jauch, J.M., Piron, C.: On the structure of quantum proposition systems. Helv. Phys. Acta42, 827–837 (1969) · Zbl 0129.41902
[10] Pedersen, G.K.:C *-algebras and their Authomorphisms Groups. New York London: Academis Press 1979 · Zbl 0416.46043
[11] Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics. Dordrecht Boston London, Kluwer 1991 · Zbl 0743.03039
[12] Mackey, G.W.: Mathematical Foundations of Quantum Mechanics. New York: Benjamin 1963 · Zbl 0114.44002
[13] Ruttimann, G.: Jauch-Piron states. J. Math. Phys.18 (2), 189–193 (1977) · Zbl 0388.03025 · doi:10.1063/1.523255
[14] Stratilla, S., Zsido, L.: Lectures on von Neumann Algebras. Tunbridge Wells: Abacus Press 1979
[15] Takesaki, M.: Theory of Operator Algebras I. Berlin Heidelberg New York: Springer 1979 · Zbl 0436.46043
[16] Yeadon, F.J.: Finitely additive measures on projections in finiteW *-algebras. Bull. Lond. Math. Soc.16, 145–150 (1984) · Zbl 0574.46048 · doi:10.1112/blms/16.2.145
[17] Zierler, N.: Axioms for non relativistic quantum mechanics. Pac. J. Math.11, 1151–1169 (1961) · Zbl 0138.44503
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