Normality and fixed points associated to commutative row contractions

Abstract

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