A refined nc Oka–Weil theorem

Abstract

This short note refines a noncommutative (nc) Oka–Weil theorem by using a characterization of free compact nc sets based on the notion of dilation hulls. A consequence of it is that any free holomorphic function can be represented as a free polynomial on each free compact nc set.

A polynomial approximation result for free Herglotz-Agler functions

“…We will also use it to show that the regular nc Schur-Agler class and the regular free Herglotz class are homeomorphic to each other via the Cayley transforms. Hence we can translate the previous polynomial approximation result for nc Schur-Agler functions [12,Theorem 3.3] into the regular free Herglotz class. In this way, we will establish the main result of this paper.…”

“…([12, Theorem 3.3]) A graded function f on B δ belongs to SA(B δ ) if and only if there exists a sequence of free polynomials {p n } ∞ n=1 such that p n converges to f uniformly on each free compact subset of B δ , and the norm of p n is uniformly less than one.…”

“…(1) The sequence in Theorem 4.1 apparently converges on each K δ,r . It is appropriate to consider this convergence because every closed polynomial polyhedron is not free compact (see the discussion below[12, Corollary 3.2]). (2) By construction, if δ(0) = 0 and f (0) = 0, then the polynomials p n in Theorem 4.1 must satisfy p n (0) = 0.…”

A polynomial approximation result for free Herglotz–Agler functions

“…The consequence is that any regular free Herglotz-Agler functions on the polynomial polyhedron B δ associated with a matrix δ of free polynomials that satisfies δ(0) = 0 can uniformly be approximated by regular Herglotz-Agler free polynomials on each K δ,r = {x ∈ B δ | ∥δ(x)∥ ≤ r} with 0 < r < 1. We have known a polynomial approximation result for nc Schur-Agler functions [12,Theorem 3.3], which assert that every regular nc 666 K. Kojin Schur-Agler function can uniformly be approximated by regular nc Schur-Agler free polynomials on each K δ,r . Hence, it is natural to translate the approximation sequence {p n } ∞ n=1 into the regular free Herglotz-Agler class via the Cayley transform f ↦ 1 + f 1 − f .…”

“…([12, Theorem 3.3]) A graded function f on B δ belongs to SA(B δ ) if and only if there exists a sequence of free polynomials {p n } ∞ n=1 such that p n converges to f uniformly on each free compact subset of B δ , and the norm of p n is uniformly less than one.Remark 4.2. (1) The sequence in Theorem 4.1 apparently converges on each K δ,r .…”