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Unbounded Negative Definite Functions

  • Charles A. Akemann (a1) and Martin E. Walter (a2)

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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 QG there is a continuous, negative definite function on G which is unbounded on Q.

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References

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1. Akemann, C. A. and Walter, M. E., The Riemann-Lebesgue property for arbitrary locally compact groups, Duke Math. J. 43 (1976), 225236.
2. Berg, C. and Forst, G., Potential theory on locally compact abelian groups, Ergebnine der Mathematik unde ihrer Grenzgebiete 87 (Springer-Verlag, Berlin, 1975).
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).
4. Dixmier, J., C*-algebras (North-Holland, 1977).
5. Haagerup, U., An example of a non-nuclear C*-algebra, which has the metric approximation property (with addendum), Invent. Math. 50 (1978-79) 279293.
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), 6365.
7. T-Y, Lee, Ph.D. Thesis, University of California at Santa Barbara (1979).
8. Paschke, W., Inner amenability and conjugation operators, Proc. Amer. Math. Soc. 71 (1978), 117118.
9. Pedersen, G. K., C*-algebras and their automorphism groups (Academic Press, 1979).
10. Sakai, S., C*-algebras and W*-algebras (Springer-Verlag, Berlin, 1971).
11. Walter, M. E., Differentiation on the dual of a group, An introduction, submitted.
12. Wang, S. P., On isolated points in the dual spaces of locally compact groups, Math. Ann. 218 (1975), 1934.
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Unbounded Negative Definite Functions

  • Charles A. Akemann (a1) and Martin E. Walter (a2)

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