Unitary invariants on the unit ball of B(H)^n

TL;DR

This paper introduces a new unitary invariant for the unit ball of B(H)^n, enabling classification of certain row contractions and analyzing the automorphism group with a concrete representation.

Contribution

It defines the invariant $$ using characteristic functions and Poisson kernels, and applies it to classify pure row isometries and contractions, also characterizing the automorphism group.

Findings

$$ detects pure row isometries

Classifies pure row contractions with polynomial characteristic functions

Describes the automorphism group as a metrizable, locally compact group

Abstract

In this paper, we introduce a unitary invariant $\Gamma$ defined on the unit ball of $B(H)^n$ in terms of the characteristic function, the noncommutative Poisson kernel, and the defect operator associated with a row contraction. We show that $\Gamma$ detects the pure row isometries and completely classify them up to a unitary equivalence. We also show that $\Gamma$ detects the pure row contractions with polynomial characteristic functions and completely non-coisometric row contractions. In particular, we show that any completely non-coisometric row contraction with constant characteristic function is homogeneous. Under a natural topology, we prove that the free holomorphic automorphism group of the unit ball of $B(H)^n$ is a metrizable, $\sigma$-compact, locally compact group, and provide a concrete unitary projective representation of it in terms of noncommutative Poisson kernels.