Extensions of Positive Definite Functions on Amenable Groups

Abstract

AbstractLet S be a subset of an amenable group G such that e ∈ 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.