Normality and fixed points associated to commutative row contractions

Let A = { A k } k = 1 n ( n is a positive integer or ∞ ) be a commutative row contraction on a complex Hilbert space H and Φ 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 ∑ k = 1 n A k ∗ A k = I . Moreover, the fixed point set B ( H ) Φ A of Φ A is considered when Φ A j ( I ) is convergent to a projection in strong operator topology.