Pure eigenstates for the sum of generators of the free group

Abstract

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.