Extensions of Positive Definite Functions on Amenable and on Free Groups (Part 1)

In the beginning, generalities about the theory of positive definite extensions of partially definite matrices are presented. Undirected graphs are associated with such matrices and the importance of chordal graphs in this context is underlined. The results are later used for proving extension results for positive definite functions on groups. A group for which there exists a left invariant mean is called amenable. If S is a symmetric subset of a group G, then the Cayley graph Gamma of G relative to S has G as its vertex set and {x,y} is an edge if the inverse of x multiplied by y is in S. We prove that in case G is amenable and Gamma is chordal, 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. In the last part, we show that every positive definite operator-valued function on words of length less or equal to m of the free group with n generators can be extended to a positive definite function on the whole group. Some related results will be presented, including factorization of positive polynomials in noncommutative variables.