  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.
