Free holomorphic automorphisms of the unit ball of $B(H)^n$

TL;DR

This paper characterizes the automorphism group of the noncommutative unit ball of operators, explores their impact on operator invariants, and extends classical complex analysis principles to a noncommutative multivariable context.

Contribution

It provides a complete description of free holomorphic automorphisms of the noncommutative unit ball and links these to automorphisms of associated operator algebras, extending classical results.

Findings

Automorphisms commute with the noncommutative Poisson transform.

Characterization of unitarily implemented automorphisms of the Cuntz-Toeplitz algebra.

Extension of maximum principle and Schwarz lemma to noncommutative setting.

Abstract

The theory of characteristic functions for row contractions is used to determine the group $Aut(B(H)^n_1)$ of all free holomorphic automorphisms of the unit ball of $B(H)^n$. We show that the noncommutative Poisson transform commutes with the action of the automorphism group $Aut(B(H)^n_1)$. This leads to a characterization of the unitarily implemented automorphisms of the Cuntz-Toeplitz algebra $C^*(S_1,..., S_n)$, which leave invariant the noncommutative disc algebra $\cA_n$. This result provides new insight into Voiculescu's group of automorphisms of the Cuntz-Toeplitz algebra and reveals new connections with noncommutative multivariable operator theory, especially, the theory of characteristic functions for row contractions and the noncommutative Poisson transforms. We study the isometric dilations and the characteristic functions of row contractions under the action of the automorphism group $Aut(B(H)^n_1)$. This enables us to obtain some results concerning the behavior of the curvature and the Euler characteristic of a row contraction under $Aut(B(H)^n_1)$. We prove a maximum principle for free holomorphic functions on the noncommutative ball $[B(H)^n]_1$ and provide some extensions of the classical Schwarz lemma to our noncommutative setting.