Trans. Amer. Math. Soc. 365 (2013), no. 12, 6243–6267
In this paper, we introduce a unitary invariant Γ 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 Γ detects the pure row isometries and completely classify them up to a unitary equivalence. We also show that Γ 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, σ-compact, locally compact group, and provide a concrete unitary projective representation of it in terms of noncommutative Poisson kernels.