Orthomodularity in infinite dimensions; a theorem of M. Solèr

Holland, Samuel S.

doi:10.1090/S0273-0979-1995-00593-8

Orthomodularity in infinite dimensions; a theorem of M. Solèr
HTML articles powered by AMS MathViewer

by Samuel S. Holland PDF

Bull. Amer. Math. Soc. 32 (1995), 205-234 Request permission

Abstract:

Maria Pia Solèr has recently proved that an orthomodular form that has an infinite orthonormal sequence is real, complex, or quaternionic Hilbert space. This paper provides an exposition of her result, and describes its consequences for Baer ${\ast }$-rings, infinite-dimensional projective geometries, orthomodular lattices, and Mackey’s quantum logic.

References

E. Artin, Geometric algebra, Interscience Publishers, Inc., New York-London, 1957. MR 0082463
Ichiro Amemiya and Huzihiro Araki, A remark on Piron’s paper, Publ. Res. Inst. Math. Sci. Ser. A 2 (1966/1967), 423–427. MR 0213266, DOI 10.2977/prims/1195195769
Reinhold Baer, Linear algebra and projective geometry, Academic Press, Inc., New York, N.Y., 1952. MR 0052795
Sterling K. Berberian, Baer *-rings, Die Grundlehren der mathematischen Wissenschaften, Band 195, Springer-Verlag, New York-Berlin, 1972. MR 0429975, DOI 10.1007/978-3-642-15071-5
Enrico G. Beltrametti and Gianni Cassinelli, The logic of quantum mechanics, Encyclopedia of Mathematics and its Applications, vol. 15, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by Peter A. Carruthers. MR 635780
Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
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

The principles of quantum mechanics

Jacques Dixmier, Position relative de deux variétés linéaires fermées dans un espace de Hilbert, Revue Sci. 86 (1948), 387–399 (French). MR 29095

The character of physical law

P. A. Fillmore and D. M. Topping, Operator algebras generated by projections, Duke Math. J. 34 (1967), 333–336. MR 209855, DOI 10.1215/S0012-7094-67-03436-9
Herbert Gross, Quadratic forms in infinite-dimensional vector spaces, Progress in Mathematics, vol. 1, Birkhäuser, Boston, Mass., 1979. MR 537283
Herbert Gross, Hilbert lattices: new results and unsolved problems, Found. Phys. 20 (1990), no. 5, 529–559. MR 1060621, DOI 10.1007/BF01883238
K. W. Gruenberg and A. J. Weir, Linear geometry, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0223375
Samuel S. Holland Jr., The current interest in orthomodular lattices, Trends in Lattice Theory (Sympos., U.S. Naval Academy, Annapolis, Md., 1966) Van Nostrand Reinhold, New York, 1970, pp. 41–126. MR 0272688
Samuel S. Holland Jr., Remarks on type $\textrm {I}$ Baer and Baer $^{\ast }$-rings, J. Algebra 27 (1973), 516–522. MR 330216, DOI 10.1016/0021-8693(73)90061-6
Samuel S. Holland Jr., $^{\ast }$-valuations and ordered $^{\ast }$-fields, Trans. Amer. Math. Soc. 262 (1980), no. 1, 219–243. MR 583853, DOI 10.1090/S0002-9947-1980-0583853-8
Samuel S. Holland Jr., Projections algebraically generate the bounded operators on real or quaternionic Hilbert space, Proc. Amer. Math. Soc. 123 (1995), no. 11, 3361–3362. MR 1273495, DOI 10.1090/S0002-9939-1995-1273495-X
Samuel S. Holland Jr., Applied analysis by the Hilbert space method, Monographs and Textbooks in Pure and Applied Mathematics, vol. 137, Marcel Dekker, Inc., New York, 1990. An introduction with applications to the wave, heat, and Schrödinger equations. MR 1084593
Josef M. Jauch, Foundations of quantum mechanics, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1968. MR 0218062
G. Kalmbach, Orthomodulare Verbände, Jahresber. Deutsch. Math.-Verein. 85 (1983), no. 1, 33–49 (German). MR 689129
Irving Kaplansky, Forms in infinite-dimensional spaces, An. Acad. Brasil. Ci. 22 (1950), 1–17. MR 37285
Irving Kaplansky, Rings of operators, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0244778
Hans Arwed Keller, Ein nicht-klassischer Hilbertscher Raum, Math. Z. 172 (1980), no. 1, 41–49 (German). MR 576294, DOI 10.1007/BF01182777

Orthomodular spaces

Mathematical foundation of quantum mechanics

F. Maeda and S. Maeda, Theory of symmetric lattices, Die Grundlehren der mathematischen Wissenschaften, Band 173, Springer-Verlag, New York-Berlin, 1970. MR 0282889, DOI 10.1007/978-3-642-46248-1

Notes on axioms for quantum mechanics

Ronald P. Morash, Angle bisection and orthoautomorphisms in Hilbert lattices, Canadian J. Math. 25 (1973), 261–272. MR 313143, DOI 10.4153/CJM-1973-026-2
C. Piron, Axiomatique quantique, Helv. Phys. Acta 37 (1964), 439–468 (French, with English summary). MR 204048
Alexander Prestel, On Solèr’s characterization of Hilbert spaces, Manuscripta Math. 86 (1995), no. 2, 225–238. MR 1317747, DOI 10.1007/BF02567991
M. P. Solèr, Characterization of Hilbert spaces by orthomodular spaces, Comm. Algebra 23 (1995), no. 1, 219–243. MR 1311786, DOI 10.1080/00927879508825218

Projective geometry

John von Neumann, Continuous geometry, Princeton Mathematical Series, No. 25, Princeton University Press, Princeton, N.J., 1960. Foreword by Israel Halperin. MR 0120174
John von Neumann, Mathematical foundations of quantum mechanics, Princeton University Press, Princeton, N.J., 1955. Translated by Robert T. Beyer. MR 0066944
W. John Wilbur, On characterizing the standard quantum logics, Trans. Amer. Math. Soc. 233 (1977), 265–282. MR 468710, DOI 10.1090/S0002-9947-1977-0468710-X
Neal Zierler, Axioms for non-relativistic quantum mechanics, Pacific J. Math. 11 (1961), 1151–1169. MR 140972, DOI 10.2140/pjm.1961.11.1151