Fixed points of normal completely positive maps on B(H)

Given a sequence of bounded operators aj on a Hilbert space H with ∑j=1∞aj⁎aj=1=∑j=1∞ajaj⁎, we study the map Ψ defined on B(H) by Ψ(x)=∑j=1∞aj⁎xaj and its restriction Φ to the Hilbert-Schmidt class C2(H). In the case when the sum ∑j=1∞aj⁎aj is norm-convergent we show in particular that the operator Φ−1 is not invertible if and only if the C⁎-algebra A generated by {aj}j=1∞ has an amenable trace. This is used to show that Ψ may have fixed points in B(H) which are not in the commutant A′ of A even in the case when the weak* closure of A is injective. However, if A is abelian, then all fixed points of Ψ are in A′ even if the operators aj are not positive.