Normality and fixed points associated to commutative row contractions

Let A={Ak}k=1n (nn is a positive integer or ∞∞) be a commutative row contraction on a complex Hilbert space HH and ΦAΦA the normal completely positive map associated with AA. We give some characterizations for AA to be a normal sequence. In the case that AA is unital, we show AA is normal if either AA is contained in a finite von Neumann algebra or the set K(H)K(H) of all compact operators or ∑k=1nAk∗Ak=I. Moreover, the fixed point set B(H)ΦAB(H)ΦA of ΦAΦA is considered when ΦAj(I) is convergent to a projection in strong operator topology.