Hyperbolic geometry on noncommutative polyballs

TL;DR

This paper introduces a hyperbolic geometric framework for noncommutative polyballs using free pluriharmonic functions and Poisson kernels, establishing hyperbolic metrics and automorphism invariance, thus viewing polyballs as noncommutative hyperbolic spaces.

Contribution

It develops a new hyperbolic metric on noncommutative polyballs and proves its properties, including invariance and completeness, extending classical hyperbolic geometry to a noncommutative operator setting.

Findings

Defined hyperbolic metrics on noncommutative polyballs.

Proved hyperbolic Schwarz-Pick lemma for free holomorphic functions.

Established invariance and completeness of the hyperbolic metric.

Abstract

This paper is an introduction to the hyperbolic geometry of noncommutative polyballs B_n of bounded linear operators on Hilbert spaces. We use the theory of free pluriharmonic functions on polyballs and noncommutative Poisson kernels on tensor products of full Fock spaces to define hyperbolic type metrics on B_n, study their properties, and obtain hyperbolic versions of Schwarz-Pick lemma for free holomorphic functions on polyballs. As a consequence, the polyballs can be viewed as noncommutative hyperbolic spaces. When specialized to the operatorial polydisk D_k, our hyperbolic metric is complete and invariant under the group of all free holomorphic automorphisms of D_k, and the topology induced on D_k is the usual operator norm topology.