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Description: |
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.
The talk is based on joint work with Dan Timotin (Mathematics Institute of the
Romanian Academy).
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