Trans. Amer. Math. Soc. 365 (2013), no. 12, 6243–6267
In this paper, we introduce a unitary invariant
defined in terms of the characteristic function , the noncommutative Poisson kernel , and the defect operator associated with . We show that the map detects the pure row isometries in the closed unit ball of and completely classify them up to a unitary equivalence. We also show that detects the pure row contractions with polynomial characteristic functions and completely noncoisometric row contractions, while the pair is a complete unitary invariant for these classes of row contractions.
The unitary invariant 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 with constant characteristic function is homogeneous, i.e., is unitarily equivalent to for any free holomorphic automorphism of the unit ball of .
Under a natural topology, we prove that the free holomorphic automorphism group is a metrizable, -compact, locally compact group, and provide a concrete unitary projective representation of it in terms of noncommutative Poisson kernels.