A polynomial approximation result for free Herglotz–Agler functions

Abstract

In this paper, we prove a noncommutative (nc for short) analog of Schwarz lemma for the nc Schur-Agler class and prove that the regular nc Schur-Agler class and the regular free Herglotz-Agler class are homeomorphic. Moreover, we give a characterization of regular free Herglotz-Agler functions. As an application, we will show that any regular free Herglotz-Agler functions can uniformly be approximated by regular Herglotz-Agler free polynomials.