Noncommutative transforms and free pluriharmonic functions

TL;DR

This paper introduces noncommutative transforms generalizing classical complex analysis tools to study free pluriharmonic functions on noncommutative balls, analyzing their boundary behavior and establishing free analogues of classical results.

Contribution

It introduces new noncommutative transforms and explores their application to free pluriharmonic functions, extending classical complex analysis results to a noncommutative multivariable context.

Findings

Development of noncommutative transforms generalizing classical ones

Analysis of boundary behavior of free pluriharmonic functions

Establishment of free analogues of classical complex analysis results

Abstract

In this paper, we study free pluriharmonic functions on noncommutative balls, and their boundary behavior. The main tools used in this study are certain noncommutative transforms which are introduced in the present paper and which generalize the classical transforms of Berezin, Poisson, Fantappie, Herglotz, and Cayley. Several classical results from complex analysis have free analogues in our noncommutative multivariable setting.