Extensions of Positive Definite Functions on Amenable Groups

Abstract

Abstract. Let 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.

Positive extensions of Schur multipliers

“…As an application of our results on positive extensions of Schur multipliers, we now provide a different approach to the main result of [4]. Theorem 5.9.…”

“…Firstly, we will see in Section 5 that the problem we consider is closely related to the problem of extending partially defined positive definite functions on locally compact groups. The latter problem has been studied in a variety of contexts and there is a rich bibliography on its modern aspects as well as its connections with classical problems and applications (see [2], [3], [4], [7], [11], [19] and the references Date: 23 December 2016. therein). Since the main interest here lies in infinite groups, the passage to infinite dimensions becomes necessary.…”

“…Closely related problems about extension of Herz-Schur multipliers were recently considered in [6]. We use the results from Section 4 to give, in the case of discrete amenable groups, a different approach to the result of Bakonyi and Timotin [4] concerning positive definite extensions of partially defined functions. Our main result here (Theorem 5.6) concerns (classes of non-discrete) locally compact groups, where we formulate a sufficient condition for the existence of positive definite extensions in terms of operator approximations.…”

“…Acknowledgements We wish to thank Yemon Choi for bringing the reference [4] to our attention. We also thank the anonymous referee for the careful reading of the manuscript and his/her useful suggestions.…”