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# Extensions of Positive Definite Functions on Amenable Groups

Published:2010-08-03
Printed: Mar 2011
• M. Bakonyi,
Department of Mathematics and Statistics, Georgia State University, Atlanta, GA, U.S.A
• D. Timotin,
Institute of Mathematics of the Romanian Academy, Bucharest, Romania
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## Abstract

Let $S$ be a subset of an amenable group $G$ such that $e\in S$ and $S^{-1}=S$. The main result of this paper states that if the Cayley graph of $G$ with respect to $S$ has a certain combinatorial property, then every positive definite operator-valued function on $S$ can be extended to a positive definite function on $G$. Several known extension results are obtained as corollaries. New applications are also presented.
 MSC Classifications: 43A35 - Positive definite functions on groups, semigroups, etc. 47A57 - Operator methods in interpolation, moment and extension problems [See also 30E05, 42A70, 42A82, 44A60] 20E05 - Free nonabelian groups