Suppose $L_n$ is a sequence of normal completely positive mappings of $B(H)$ into itself of norm strictly less than one with the property that $L_n(L_m(A)) \to L_m(A)$ in the strong operator topology as $n \to \infty$. For $A$ in $B(H)$ let $$ Q_n(A) = A + L_n(A) + L_n^2(A) + L_n^3(A) + \ldots $$ (The series converges in norm since $L_n$ has norm less than one.) You are given that the $Q_n$ are increasing in that $Q_n - Q_m$ is completely positive for $n > m$
Suppose $L$ is a limit point of the $L_n$ and the range of $L$ is a factor $M$. Is $M$ a type $\mathrm{I}$ factor?
I have an example of such a limit where the range of $L$ is $L^\infty(\Bbb R)$ (bounded measurable functions on the real line) so the range need not be atomic. (The $L_n$ come from my theory of generalized boundary representation of type $\mathrm{II}_0$ $E-0$-semigroups so they have further, complicated properties I don't really understand but I am hoping I have given enough properties so that one can come to a conclusion about the limiting factor without going into them). Thanks for a speedy answer to my first question.
