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

Popescu, Gelu

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.