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

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.