Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Lattice properties of subspace families in an inner product space

Authors: Pavel Pták and Hans Weber
Journal: Proc. Amer. Math. Soc. 129 (2001), 2111-2117
MSC (2000): Primary 03G12, 46C05, 81P10
Published electronically: February 15, 2001
MathSciNet review: 1825924
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information


Let $S$ be a separable inner product space over the field of real numbers. Let $E(S)$ (resp., $C(S))$ denote the orthomodular poset of all splitting subspaces (resp., complete-cocomplete subspaces) of $S$. We ask whether $E(S)$ (resp., $C(S))$ can be a lattice without $S$ being complete (i.e. without $S$ being Hilbert). This question is relevant to the recent study of the algebraic properties of splitting subspaces and to the search for ``nonstandard'' orthomodular spaces as motivated by quantum theories. We first exhibit such a space $S$ that $E(S)$ is not a lattice and $C(S)$ is a (modular) lattice. We then go on showing that the orthomodular poset $E(S)$ may not be a lattice even if $E(S)=C(S)$. Finally, we construct a noncomplete space $S$ such that $E(S)=C(S)$ with $E(S)$ being a (modular) lattice. (Thus, the lattice properties of $E(S)$ (resp. $C(S))$ do not seem to have an explicit relation to the completeness of $S$ though the Ammemia-Araki theorem may suggest the opposite.) As a by-product of our construction we find that there is a noncomplete $S$ such that all states on $E(S)$ are restrictions of the states on $E(\overline{S})$ for $\overline{S}$ being the completion of $S$ (this provides a solution to a recently formulated problem).

References [Enhancements On Off] (What's this?)

  • 1. H. Araki Amemiya, A remark on Piron's paper, Publ. Res. Inst. Math. Sci., Kyoto Univ., Ser. A 2 (1966), 423-427. MR 35:4130
  • 2. A. Dvurecenskij, Gleason's Theorem and Applications, Kluwer Academic Publishers, Dordrecht/Boston/London, 1993. MR 94m:46092
  • 3. A. Dvurecenskij, S. Pulmannová, States on splitting subspaces and completeness of inner product spaces, Inter. J. Theor. Phys. 27 (1988), 1059-1067. MR 90b:46050
  • 4. H. Gross, H. Keller, On the definition of Hilbert space, Manuscr. Math. 23 (1977), 67-90. MR 58:2188
  • 5. P. Halmos, A Hilbert Space Problem Book, Van Nostrand-Reinhold, Princeton, New Jersey, 1967. MR 34:8178
  • 6. J. Hamhalter, P. Pták, A completeness criterion for inner product spaces, Bull. London Math. Soc. 19 (1987), 259-263. MR 88a:46019
  • 7. J. Harding, Orthomodular lattices whose MacNeille completions are not orthomodular, Order 8 (1991), 93-103. MR 92k:06015
  • 8. J. Harding, Decompositions in quantum logic, Trans. Amer. Math. Soc., Vol. 348, 5 (1996), 1839-1862. MR 96h:81003
  • 9. S. S. Holland, Jr., Orthomodularity in infinite dimensions; a theorem of M. Solér, Bull. Amer. Math. Soc., Vol. 32, No. 2 (1995), 205-234. MR 95j:06010
  • 10. G. Kalmbach, Measures and Hilbert Lattices, World Sci. Publ. Co., Singapore, 1986. MR 88a:06013
  • 11. H. A. Keller, Ein nicht-klassischer Hilbertischen Raum, Math. Z. 272 (1980), 42-49. MR 81f:46033
  • 12. F. Maeda, S. Maeda, Theory of Symmetric Lattices, Springer-Verlag, Berlin, 1970. MR 44:123
  • 13. R. Mayet, Some characterizations of the underlying division ring of a Hilbert lattice by automorphisms, Int. J. Theor. Phys. 37 (1998), 109-114. MR 99f:81020
  • 14. R. Piziak, Lattice theory, quadratic spaces, and quantum proposition systems, Found. Phys. 20 (1990), 651-665. MR 91h:06021
  • 15. P. Pták, S. Pulmannová, Orthomodular Structures as Quantum Logics, Kluwer, 1991. MR 94d:81018b
  • 16. M. P. Solér, Characterization of Hilbert spaces by orthomodular spaces, Comm. in Algebra 23 (1) (1995), 219-243. MR 95k:46035
  • 17. V. Varadarajan, Geometry of Quantum Theory I, II, Van Nostrand, Princeton, 1968, 1970. MR 57:11399; MR 57:11400

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 03G12, 46C05, 81P10

Retrieve articles in all journals with MSC (2000): 03G12, 46C05, 81P10

Additional Information

Pavel Pták
Affiliation: Faculty of Electrical Engineering, Department of Mathematics, Czech Technical University, 166 27 Prague 6, Czech Republic

Hans Weber
Affiliation: Dipartimento di Matematica, e Informatica, Università degli Studi di Udine, I-33100 Udine, Italy

Keywords: Inner product (= prehilbert) space, splitting subspace, orthomodular poset, lattice.
Received by editor(s): September 30, 1998
Received by editor(s) in revised form: June 2, 1999
Published electronically: February 15, 2001
Additional Notes: The authors acknowledge the support of grant GAČR 201/98/1153 of the Czech Grant Agency and Progetto di ricerca di interesse nazionale Analisi Reale (Italy).
Communicated by: David R. Larson
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society