Toeplitz Corona and the Douglas property for free functions

The well known Douglas Lemma says that for operators A, B   on Hilbert space that AA⁎−BB⁎⪰0AA⁎−BB⁎⪰0 implies B=ACB=AC for some contraction operator C. The result carries over directly to classical operator-valued Toeplitz operators simply by replacement of operator by Toeplitz operator throughout. Free functions generalize the notion of free polynomials and formal power series and trace back to the work of J. Taylor in the 1970s. They are of current interest, in part because of their connections with free probability and engineering systems theory. In this article, for given free functions a and b   with noncommutative domain KK defined by free polynomial inequalities, we obtain a sufficient condition in terms of the difference aa⁎−bb⁎aa⁎−bb⁎ for the existence of a free function c   taking contractive values on KK such that b=acb=ac. The connection to recent work of Agler and McCarthy and their free Toeplitz Corona Theorem is expounded.