We consider certain
positive definite functions on a finitely generated free group G that are defined
with respect to a given basis in terms of word length and the number of
negative-to-positive generator exponent switches. Some of these functions are
eigenfunctions for right convolution by the sum of the generators, and give rise
to irreducible unitary representations of G. We show that any state of the
reduced C*-algebra of G whose left kernel contains a polynomial in one
of the generators must factor through the conditional expectation on the
C*-subalgebra generated by that generator. Our results lend some support to the
conjecture that an element of the complex group algebra of G can lie in the
left kernel of only finitely many pure states of the reduced C*-algebra of
G.
