Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T03:21:29.707Z Has data issue: false hasContentIssue false
  • English
  • Français

Topological aspects of the projective unitary group

Published online by Cambridge University Press:  24 October 2008

D. J. Simms
D. J. Simms
Affiliation:
Trinity College, Dublin
Get access Rights & Permissions [Opens in a new window]

Extract

1. Introduction. The group U(H) of unitary transformations of a complex Hilbert space H, endowed with its strong operator topology, is of interest in the study of unitary representations of a topological group. The unitary transformations of H induce a group U(Ĥ) of transformations of the associated projective space Ĥ. The projective unitary group U(Ĥ) with its strong operator topology is used in the study of projective (ray) representations. U(Ĥ) is, as a group, the quotient of U(H) by the subgroup S1 of scalar multiples of the identity. In this paper we prove that the strong operator toplogy of U(Ĥ) is in fact the quotient of the strong operator topology on U(H). This is related to the fact that U(H) is a principal bundle over U(Ĥ) with fibre S.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 68 , Issue 1 , July 1970 , pp. 57 - 60
Copyright
Copyright © Cambridge Philosophical Society 1970

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Bargmann, V.On unitary ray representations of continuous groups. Ann. of Math. 59 (1954), 1–46.CrossRefGoogle Scholar
(2)Cartan, É.La topologie des espaces représentatifs des groupes de Lie. Oeuvres Complètes, Partie 1, Vol. 2, Num. 150.Google Scholar
(3)Simms, D. J.Lie Groups and Quantum Mechanics. Lecture notes in mathematics, Vol. 52 (Springer, 1968).Google Scholar
(4)Spanier, E. H.Algebraic topology (McGraw-Hill, 1966).Google Scholar
(5)Steenrod, N.The topology of fibre bundles (Princeton, 1951).CrossRefGoogle Scholar
(6)Wigner, E. P.On unitary representations of the inhomogeneous Lorentz group. Ann. of Math. 40 (1939), 149–204.Google Scholar