Free pluriharmonic majorants and noncommutative interpolation

TL;DR

This paper explores noncommutative pluriharmonic functions, characterizes a noncommutative Hardy space, and solves a multivariable lifting problem using free pluriharmonic majorants.

Contribution

It introduces the study of sub-pluriharmonic curves in Cuntz-Toeplitz algebras and characterizes the noncommutative Hardy space via free pluriharmonic majorants.

Findings

Characterization of the noncommutative Hardy space $H^2_{\bf ball}$

Schur type description of the unit ball of $H^2_{\bf ball}$

Solution to a multivariable commutant lifting problem

Abstract

In this paper, we initiate the study of sub-pluriharmonic curves in Cuntz-Toeplitz algebras and free pluriharmonic majorants on noncommutative balls. We are lead to a characterization of the noncommutative Hardy space $H^2_{\bf ball}$ in terms of free pluriharmonic majorants, and to a Schur type description of the unit ball of $H^2_{\bf ball}$. These results are used to solve a multivariable commutant lifting problem and provide a description of all solutions.