Suppose L_n$L_n$ is a sequence of normal completely positive mappings of B(H)$B(H)$ into itself of norm strictly less than one with the property that L_n(L_m(A)) -> L_m(A)$L_n(L_m(A)) \to L_m(A)$ in the strong operator topology as n -> infinity$n \to \infty$. For A$A$ in B(H) let
Q_n(A) = A + L_n(A) + L_n^2(A) + L_n^3(A) + ....
$B(H)$ 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$L_n$ has norm less than one.) You are given that that the Q_n$Q_n$ are increasing in that Q_n - Q_m$Q_n - Q_m$ 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$n > m$
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 factor. I have an example of such a limit where the range of L$L$ is L^infinity(R)$L^\infty(\Bbb R)$ (bounded measurable functions on the real line) so the range need not be atomic. (The L_n$L_n$ come from my theory of generalized boundary representation of type II_0 E-0$\mathrm{II}_0$ $E-0$-semigroups so they have further properties complicated, 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.
