|
Only `free' measures are admissable on  when the inner product space  is incomplete
Author(s):
D.
Buhagiar;
E.
Chetcuti
Journal:
Proc. Amer. Math. Soc.
136
(2008),
919-922.
MSC (2000):
Primary 46C05, 46C15;
Secondary 46L30
Posted:
November 30, 2007
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
Using elementary arguments and without having to recall the Gleason Theorem, we prove that the existence of a nonsingular measure on the lattice of orthogonally closed subspaces of an inner product space  is a sufficient (and of course, a necessary) condition for  to be a Hilbert space.
References:
-
- 1.
- I. Amemiya and H. Araki, A remark on Piron's paper, Publ. Res. Inst. Math. Sci., Kyoto Univ. Ser. A 2 (1966-67), 423-427. MR 0213266 (35:4130)
- 2.
- E. Chetcuti and A. Dvurečenskij, A finitely additive state criterion for the completeness of inner product spaces, Letters Math. Phys. 64 (2003), 221-227. MR 2009260 (2004j:46029)
- 3.
- -, The state-space of the lattice of orthogonally closed subspaces, Glasgow Math. J. 47 (2005), 213-220. MR 2200969 (2006i:46034)
- 4.
- A. Dvurečenskij, Gleason's Theorem and Its Applications, Kluwer Acad. Publ., Dordrecht, Ister Science Press, Bratislava, 1992.
- 5.
- A. Dvurečenskij, T. Neubrunn and S. Pulmannová, Finitely additive states and completeness of inner product spaces, Found. Phys. 20 (1990), 1091-1102. MR 1078957 (92a:46026)
- 6.
- A. M. Gleason, Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6 (1957), 885-893. MR 0096113 (20:2609)
- 7.
- S. P. Gudder, Inner product spaces, Amer. Math. Monthly 82 (1975), 251-252. MR 0420234 (54:8248)
- 8.
- J. Hamhalter and P. Pták, A completeness criterion for inner product spaces, Bull. London Math. Soc. 19 (1987), 259-263. MR 879514 (88a:46019)
- 9.
- J. Hamhalter, Quantum Measure Theory, Kluwer Academic Publishers, Dordrecht, Boston, London, 2003. MR 2015280 (2004j:46085)
- 10.
- F. Maeda and S. Maeda, Theory of Symmetric Lattices, Springer-Verlag, Berlin, 1970. MR 0282889 (44:123)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
46C05, 46C15,
46L30
Retrieve articles in all Journals with MSC
(2000):
46C05, 46C15,
46L30
Additional Information:
D.
Buhagiar
Affiliation:
Department of Mathematics, Faculty of Science, University of Malta, Msida MSD.06, Malta
Email:
david.buhagiar@um.edu.mt
E.
Chetcuti
Affiliation:
Department of Mathematics, Junior College, University of Malta, Msida MSD.06, Malta
Email:
emanuel.chetcuti@um.edu.mt
DOI:
10.1090/S0002-9939-07-08982-4
PII:
S 0002-9939(07)08982-4
Received by editor(s):
May 24, 2006
Received by editor(s) in revised form:
October 11, 2006
Posted:
November 30, 2007
Communicated by:
David Preiss
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org