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

DOI:
https://doi.org/10.1090/S0002-9939-01-05855-5

Published electronically:
February 15, 2001

MathSciNet review:
1825924

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Let be a separable inner product space over the field of real numbers. Let (resp., denote the orthomodular poset of all splitting subspaces (resp., complete-cocomplete subspaces) of . We ask whether (resp., can be a lattice without being complete (i.e. without 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 that is not a lattice and is a (modular) lattice. We then go on showing that the orthomodular poset may not be a lattice even if . Finally, we construct a noncomplete space such that with being a (modular) lattice. (Thus, the lattice properties of (resp. do not seem to have an explicit relation to the completeness of though the Ammemia-Araki theorem may suggest the opposite.) As a by-product of our construction we find that there is a noncomplete such that all states on are restrictions of the states on for being the completion of (this provides a solution to a recently formulated problem).

**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**

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

Email:
ptak@math.feld.cvut.cz

**Hans Weber**

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

Email:
weber@dimi.uniud.it

DOI:
https://doi.org/10.1090/S0002-9939-01-05855-5

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