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

Abstract

In this paper we continue the study of free holomorphic functions on the noncommutative ball

where B(ℋ) is the algebra of all bounded linear operators on a Hilbert space ℋ, and n = 1, 2, . . . or n = ∞. These are noncommutative multivariable analogues of the analytic functions on the open unit disc 𝔻 ≔ {z ∈ ℂ : |z| < 1}.

The theory of characteristic functions for row contractions (elements in ) is used to determine the group of all free holomorphic automorphisms of [B(ℋ)ⁿ]₁. It is shown that , the Moebius group of the open unit ball 𝔹_n ≔ {λ ∈ ℂⁿ : ∥λ∥₂ < 1}.

We show that the noncommutative Poisson transform commutes with the action of the automorphism group . This leads to a characterization of the unitarily implemented automorphisms of the Cuntz-Toeplitz algebra C*(S₁, . . . , S_n), which leave invariant the noncommutative disc algebra . 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 show that the unitarily implemented automorphisms of the noncommutative disc algebra and the noncommutative analytic Toeplitz algebra , respectively, are determined by the free holomorphic automorphisms of [B(ℋ)ⁿ]₁, via the noncommutative Poisson transform. Moreover, we prove that

.

We also prove that any completely isometric automorphism of the noncommutative disc algebra has the form

where . We deduce a similar result for the noncommutative Hardy algebra , which, due to Davidson-Pitts results on the automorphisms of , implies that is isomorphic to the group of contractive automorphisms of .

We study the isometric dilations and the characteristic functions of row contractions under the action of the automorphism group . This enables us to obtain some results concerning the behavior of the curvature and the Euler characteristic of a row contraction under .

Finally, in the last section, 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.