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

By Gelu Popescu. 2008

113 citations recorded by the source corpus.

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*(S 1, . . . , Sn ), 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.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7fc52fd5ea3e83e9342032a4d6a88f8d1c7906db

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Citations in the corpus

Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients (2013, unknown): “ometric and completely isometric isomorphism. LetAut(Bd)denote thegroup of automorphisms of the ball, that is, the biholomorphisms of Bd onto itself. Theorem 11.1.1 ([49] Section 4;[51], Theorem 9.2, [103], Theorems 3.5 and 3.10). For every φ∈Aut(Bd) there exists a unitary U: H2 d →H 2 d given by (1”
Free holomorphic functions on the unit ball of B(H)n (2006, unknown): “All rights reserved. doi:10.1016/j.jfa.2009.10.014 Keywords: Multivariable operator theory; Noncommutative function theory; Free holomorphic functions; Fock space; Poisson transform; Schwarz’s lemma; Julia’s lemma; Pick’s theorem; Pseudohyperbolic metric; Borel–Carathéodory theorem; von Neumann ineq”
The implicit function theorem and free algebraic sets (2014, unknown): “ns have been studied for a variety of reasons: by Anderson [4] as a generalization of the Weyl calculus; by Taylor [20], in the context of the functional calculus for non-commuting operators; Popescu [14, 15, 16, 17], in the context of extending classical function theory to d-tuples of bounded opera”
Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball (2017, methodology): “with the commutative result [73, Section 2.2.8]. The following is a version of Cartan’s uniqueness theorem in the nc setting. This too, has been considered, from different perspectives (e.g., [48] or [69]). Let f: U!M dbe an nc holomorphic function, where Uis an nc domain. We will write f= (f 1;:::;”
Global Holomorphic Functions in Several Noncommuting Variables (2015, unknown): “This has been done in a variety of ways: by Taylor [23], in the context of the functional calculus for non-commuting operators; Voiculescu [24, 25], in the context of free probability; Popescu [18, 19, 20, 21], in the context of extending classical function theory to d-tuples Partially supported by ”
Foundations of Free Noncommutative Function Theory (2012, methodology): “[89] G. Popescu.”