Finitely additive Gleason measures

Dvurečenskij, Anatolij

doi:10.1090/S0002-9939-1992-1072335-9

Finitely additive Gleason measures
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Proc. Amer. Math. Soc. 115 (1992), 191-198 Request permission

Abstract:

We describe the set of all finitely additive measures which attain also infinite values on a quantum logic of a Hilbert space and which are expressible via the generalized Gleason-Lugovaja-Sherstnev formula. We prove that this set consists of those which are regular with respect to the set of all finite-dimensional subspaces. In addition, we show that this regularity does not entail the countable additivity, in general.

References

Johan F. Aarnes, Quasi-states on $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 149 (1970), 601–625. MR 282602, DOI 10.1090/S0002-9947-1970-0282602-4
Garrett Birkhoff and John von Neumann, The logic of quantum mechanics, Ann. of Math. (2) 37 (1936), no. 4, 823–843. MR 1503312, DOI 10.2307/1968621
Thomas Drisch, Generalization of Gleason’s theorem, Internat. J. Theoret. Phys. 18 (1979), no. 4, 239–243. MR 552434, DOI 10.1007/BF00671760
Anatolij Dvurečenskij, Generalization of Maeda’s theorem, Internat. J. Theoret. Phys. 25 (1986), no. 10, 1117–1124. MR 868372, DOI 10.1007/BF00671687
Anatolij Dvurečenskij, Note on a construction of unbounded measures on a nonseparable Hilbert space quantum logic, Ann. Inst. H. Poincaré Phys. Théor. 48 (1988), no. 4, 297–310 (English, with French summary). MR 969168
Anatolij Dvurečenskij, Solution to a regularity problem, Proceedings of the Second Winter School on Measure Theory (Lipovský Ján, 1990) Jedn. Slovensk. Mat. Fyz., Bratislava, 1990, pp. 224–227. MR 1118434
Anatolij Dvurečenskij, Tibor Neubrunn, and Sylvia Pulmannová, Regular states and countable additivity on quantum logics, Proc. Amer. Math. Soc. 114 (1992), no. 4, 931–938. MR 1045591, DOI 10.1090/S0002-9939-1992-1045591-0
Manfred Eilers and Ernst Horst, The theorem of Gleason for nonseparable Hilbert spaces, Internat. J. Theoret. Phys. 13 (1975), no. 6, 419–424. MR 389054, DOI 10.1007/BF01808324
Andrew M. Gleason, Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6 (1957), 885–893. MR 0096113, DOI 10.1512/iumj.1957.6.56050
Gudrun Kalmbach, Orthomodular lattices, London Mathematical Society Monographs, vol. 18, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1983. MR 716496

Bilinear forms generating measures on projectors

2

On a construction of unbounded measures on projectors of a Hilbert space

G. D. Lugovaja and A. N. Šerstnëv, The Gleason theorem for unbounded measures, Izv. Vyssh. Uchebn. Zaved. Mat. 12 (1980), 30–32 (Russian). MR 606673

The mathematical foundations of quantum mechanics

Lattice theory and quantum logics

New results in non-commutative measure theory

Barry Simon, A canonical decomposition for quadratic forms with applications to monotone convergence theorems, J. Functional Analysis 28 (1978), no. 3, 377–385. MR 500266, DOI 10.1016/0022-1236(78)90094-0
V. S. Varadarajan, Geometry of quantum theory. Vol. I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1968. MR 0471674