Normality and fixed points associated to commutative row contractions

Let A = {A(k)}(k=1)(n) (n is a positive integer or infinity) be a commutative row contraction on a complex Hilbert space H and Phi(A) the normal completely positive map associated with A. We give some characterizations for A to be a normal sequence. In the case that A is unital, we show A is normal if either A is contained in a finite von Neumann algebra or the set K(H) of all compact operators or Sigma(n)(k=1) A(k)*A(k)= I. Moreover, the fixed point set B (H)(Phi A) of Phi(A) is considered when Phi(j)(A) (I) is convergent to a projection in strong operator topology. (C) 2012 Elsevier Inc. All rights reserved.