William L. Paschke | Searcharxiv

arXiv2000

Some irreducible free group representations in which a linear combination of the generators has an eigenvalue

We construct irreducible unitary representations of a finitely generated free group which are weakly contained in the left regular representation and in which a given linear combination of the generators has an eigenvalue. When the eigenvalue is specified, we conjecture that there is only one such representation. The representation we have found is described explicitly (modulo inversion of a certain rational map on euclidean space) in terms of a positive definite function, and also by means of a quasi-invariant probability measure on the combinatorial boundary of the group.

math.OA↗

arXiv2000

Pure eigenstates for the sum of generators of the free group

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.

math.OA↗