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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.

Information

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

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