Eigenstates for the Sum of Generators of the Free Group
by
Bill Paschke
University of Kansas

We consider positive definite functions f on the free group on n generators of the form

f(s) = a|s|-2 c(s) bc(s)  ,

where |s| is the length of s, c(s) is the number of negative-to-positive generator exponent changes in s, and a and b are real constants with |a| at most 1, and (n-1)b between na² - 1 and n-1. In case (n-1) b = na² - 1, the state f has the sum of the generators minus na in its left kernel, and is pure; we describe the associated irreducible representation. Our results offer some support for the following conjecture: for x in the complex group algebra of the free group, only finitely many pure states of the reduced C^*-algebra can annihilate x^*x.

Date received: April 23, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacw-29.