Fixed points of normal completely positive maps on

Given a sequence of bounded operators a j on a Hilbert space H with ∑ j = 1 ∞ a j ⁎ a j = 1 = ∑ j = 1 ∞ a j a j ⁎ , we study the map Ψ defined on B ( H ) by Ψ ( x ) = ∑ j = 1 ∞ a j ⁎ x a j and its restriction Φ to the Hilbert–Schmidt class C 2 ( H ) . In the case when the sum ∑ j = 1 ∞ a j ⁎ a j 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 { a j } 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 a j are not positive.