Given a sequence of bounded operators on a Hilbert space with , we study the map Ψ defined on by and its restriction Φ to the Hilbert–Schmidt class . In the case when the sum is norm-convergent we show in particular that the operator is not invertible if and only if the -algebra A generated by has an amenable trace. This is used to show that Ψ may have fixed points in which are not in the commutant 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 even if the operators are not positive.