Hostname: page-component-68945f75b7-qvshk Total loading time: 0 Render date: 2024-08-05T16:02:58.558Z Has data issue: false hasContentIssue false
  • English
  • Français

On L1-kernels of unitary representations of semisimple Lie groups

Published online by Cambridge University Press:  24 October 2008

Mohammed E. B. Bekka  and
Jean Ludwig
Mohammed E. B. Bekka
Affiliation:
Institut de Mathématiques, Université de Lausanne, CH-1015 Lausanne, Suisse
Jean Ludwig
Affiliation:
Département de Mathématiques et Informatique, Université de Metz, Ile du Saulcy, JF-57045 Metz, France
Get access Rights & Permissions [Opens in a new window]

Extract

Let G be a locally compact group with fixed left Haar measure dx. Recall that G is said to be amenable if there exists a left translation invariant mean on the space L∞(G), i.e. if there exists a positive, linear functional M on L∞(G) such that M(lG) = 1 and M(xø) = Mø for all ø∈L∞(G), x∈G, where xø denotes the left translate xø(y) = ø(xy). The class of amenable groups includes all soluble and all compact groups (concerning the theory of amenable groups we refer to [9]). It is easy to see that G is amenable if and only if ℂ1G, the space of the constant functions on G, has a closed left translationinvariant complement in L∞(G). This reformulation of amenability leads to the following more general question.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 2 , September 1992 , pp. 343 - 348
Copyright
Copyright © Cambridge Philosophical Society 1992

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]Bekka, M. B.. Complemented subspaces of L ∞(G), ideals of L 1(G) and amenability. Monatsh. Math. 109 (1990), 195–203.CrossRefGoogle Scholar
[2]Bekka, M. B. and Ludwig, J.. Complemented *-primitive ideals of L 1-algebras of exponential Lie groups and of motion groups. Math. Z. 204 (1990), 515–526.CrossRefGoogle Scholar
[3]Dixmier, J.. C*-algebras (North-Holland, 1977).Google Scholar
[4]Gilbert, J. E.. On projections of L ∞(G) onto translation-invariant subspaces. Proc. London Math. Soc. (3) 19 (1989), 69–88.Google Scholar
[5]Greenleaf, F. P., Moskowitz, M. and Rotschild, L. P.. Central idempotent measures on connected locally compact groups. J. Funct. Anal. 15 (1974), 22–32.CrossRefGoogle Scholar
[6]Hauenschild, W. and Ludwig, J.. The injection and the projection theorem for spectral sets. Monatsh. Math. 92 (1981), 167–177.CrossRefGoogle Scholar
[7]Howe, R. and Moore, C. C.. Asymptotic properties of unitary representations. J. Funct. Anal. 32 (1979), 72–96.CrossRefGoogle Scholar
[8]Liu, T. S., van Rooij, A. and Wang, J. K.. Projections and approximate identities for ideals in group algebras. Trans. Amer. Math. Soc. 175 (1973), 469–482.CrossRefGoogle Scholar
[9]Pier, J.-P.. Amenable Locally Compact Groups (Wiley, 1984).Google Scholar
[10]Reiter, H.. L1-Algebras and Segal Algebras. Lecture Notes in Math. vol. 231 (Springer-Verlag, 1972).Google Scholar
[11]Varadarajan, V. S.. Lie Groups, Lie Algebras and their Representations (Prentice-Hall, 1974).Google Scholar
[12]Warner, G.. Harmonic Analysis on Semi-Simple Lie Groups, vol. 1 (Springer-Verlag, 1971).Google Scholar
[13]Willis, G. A.. Approximate units in finite codimensional ideals of group algebras. J. London Math. Soc. (2) 26 (1982), 143–154.CrossRefGoogle Scholar