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Unbounded Negative Definite Functions

Published online by Cambridge University Press:  20 November 2018

Charles A. Akemann  and
Martin E. Walter
Charles A. Akemann
Affiliation:
University of California, Santa Barbara, California
Martin E. Walter
Affiliation:
University of Colorado, Boulder, Colorado
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Negative definite functions (all definitions are given in § 1 below) on a locally compact, σ-compact group G have been used in several different contexts recently [2, 5, 7, 11]. In this paper we show how such functions relate to other properties such a group may have. Here are six properties which G might have. They are grouped into three pairs with one property of each pair involving negative definite functions. We show that the paired properties are equivalent and, where possible, give counter-examples to other equivalences. We assume throughout that G is not compact.

(1A) G does not have property T.

(IB) There is a continuous, negative definite function on G which is unbounded.

(2A) G has the (weak and/or strong) dual R-L property.

(2B) For every closed, non-compact set Q ⊂ G there is a continuous, negative definite function on G which is unbounded on Q.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 4 , 01 August 1981 , pp. 862 - 871
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Akemann, C. A. and Walter, M. E., The Riemann-Lebesgue property for arbitrary locally compact groups, Duke Math. J. 43 (1976), 225–236.Google Scholar
2. Berg, C. and Forst, G., Potential theory on locally compact abelian groups, Ergebnine der Mathematik unde ihrer Grenzgebiete 87 (Springer-Verlag, Berlin, 1975).Google Scholar
3. Delaroche, C. and Kirillov, A., Sur les relations entre l'espace dual d'un groupe et la structure de ses sous-groupes fermés, Séminaire Bourbaki no. 343 (1967/68).Google Scholar
4. Dixmier, J., C*-algebras (North-Holland, 1977).Google Scholar
5. Haagerup, U., An example of a non-nuclear C*-algebra, which has the metric approximation property (with addendum), Invent. Math. 50 (1978-79) 279–293.Google Scholar
6. Kazhdan, D. A., Connection of the dual space of a group with the structure of its closed subgroups, Functional Analysis and its Applications 1 (1967), 63–65.Google Scholar
7. T-Y, Lee, Ph.D. Thesis, University of California at Santa Barbara (1979).Google Scholar
8. Paschke, W., Inner amenability and conjugation operators, Proc. Amer. Math. Soc. 71 (1978), 117–118.Google Scholar
9. Pedersen, G. K., C*-algebras and their automorphism groups (Academic Press, 1979).Google Scholar
10. Sakai, S., C*-algebras and W*-algebras (Springer-Verlag, Berlin, 1971).Google Scholar
11. Walter, M. E., Differentiation on the dual of a group, An introduction, submitted.Google Scholar
12. Wang, S. P., On isolated points in the dual spaces of locally compact groups, Math. Ann. 218 (1975), 19–34.Google Scholar