Unitary invariants on the unit ball of $B(\mathcal {H})^n$

Abstract:

In this paper, we introduce a unitary invariant \[ \Gamma :[B(\mathcal {H})^n]_1^-\to \mathbb {N}_\infty \times \mathbb {N}_\infty \times \mathbb {N}_\infty ,\qquad \mathbb {N}_\infty :=\mathbb {N}\cup \{ \infty \},\] defined in terms of the characteristic function $\Theta _T$, the noncommutative Poisson kernel $K_T$, and the defect operator $\Delta _T$ associated with $T\in [B(\mathcal {H})^n]_1^-$. We show that the map $\Gamma$ detects the pure row isometries in the closed unit ball of $B(\mathcal {H})^n$ 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 noncoisometric row contractions, while the pair $(\Gamma , \Theta _T)$ is a complete unitary invariant for these classes of row contractions.

The unitary invariant $\Gamma$ is extracted from the theory of characteristic functions and noncommutative Poisson transforms, and from the geometric structure of row contractions with polynomial characteristic functions which are studied in this paper. As an application, we characterize the row contractions with constant characteristic function. In particular, we show that any completely noncoisometric row contraction $T$ with constant characteristic function is homogeneous, i.e., $T$ is unitarily equivalent to $\varphi (T)$ for any free holomorphic automorphism $\varphi$ of the unit ball of $B(\mathcal {H})^n$.

Under a natural topology, we prove that the free holomorphic automorphism group $\operatorname {Aut}(B(\mathcal {H})^n_1)$ is a metrizable, $\sigma$-compact, locally compact group, and provide a concrete unitary projective representation of it in terms of noncommutative Poisson kernels.