Operator noncommutative functions

Abstract

We establish a theory of noncommutative (NC) functions on a class of von Neumann algebras with a particular direct sum property, e.g., $B({\mathcal H})$ . In contrast to the theory’s origins, we do not rely on appealing to results from the matricial case. We prove that the $k{\mathrm {th}}$ directional derivative of any NC function at a scalar point is a k-linear homogeneous polynomial in its directions. Consequences include the fact that NC functions defined on domains containing scalar points can be uniformly approximated by free polynomials as well as realization formulas for NC functions bounded on particular sets, e.g., the NC polydisk and NC row ball.

A polynomial approximation result for free Herglotz–Agler functions

“…Salomon, Shalit and Shamovich [18,Lemma 6.11] has already proved a noncommutative analog of maximum principle for nc functions defined on unitary conjugation invariant nc domains containing 0. However, our proof is quite different from theirs, and still works even in an infinite dimensional setting like [4, chapter 16] and [7]. (We do not know how to apply their proof to the infinite dimensional setting.)…”

“…immediately follows from this expression with the aid of the Neumann series. More precisely, we have 7 One can also prove the above inequalities by using [13,Lemma 3.3].…”

“…(1) By the same calculation, we can prove an analog of Schwarz lemma for a function that admits a realization formula. Such examples are, the nc Schur-Agler class in the Ball, Marx and Vinnikov framework[8], the operator NC Schur-Agler class studied by Augat and McCarthy[7], the Schur-Agler class[1,5]…”

“…By the same calculation, we can prove an analog of Schwarz lemma for a function that admits a realization formula. Such examples are the nc Schur-Agler class in the Ball, Marx, and Vinnikov framework[8], the operator NC Schur-Agler class studied by Augat and McCarthy[7], the Schur-Agler class[1,5] (note that this class is nothing but the "level 1" of the nc Schur-Agler class; see[3, Theorem 8.19]), and the contractive multipliers of an irreducible complete Pick Hilbert function space[2]. In the last case, we have to calculate the zeros of the injection b in [2, Theorem 8.2].…”