The point-ultraweak convergence of contractive completely positive map

Question

Let $A$ and $C$ be C*-algebras. If $\phi_{n}: A \rightarrow C$ is a c.c.p (contractive completely positive) map, then the point-ultraweak cluster point of the map $\phi_{n}$ is still a c.c.p. map? Why? What about the point-norm cluster point of $\phi_{n}$?

Answer 1

The point-ultraweak topology is described in page 5 of Brown-Ozawa. Other authors (Arveson, Paulsen) call it the BW (for "bounded-weak") topology.

If $\phi_t:A\to B(H)$ are ccp, then by Theorem 1.3.7 in Brown-Ozawa it has a cluster point $\phi$. By replacing the net with an appropriate subnet, we can assume for notation simplicity that $\phi_t\to \phi$ point-ultraweakly.

Since the ultraweak topology agrees with the weak operator topology on bounded sets, it is enough to consider weak operator limits. So, given $X\in M_n(A)^+$ and $\xi\in H^n$, $$ \langle \phi^{(n)}(X)\xi,\xi\rangle=\langle[\phi(X_{kj})]\xi,\xi\rangle=\sum_{k,j=1}^n\langle\phi(X_{kj})\xi_j,\xi_k\rangle=\lim_t\sum_{k,j=1}^n\langle\phi_t(X_{kj})\xi_j,\xi_k\rangle =\lim_t\langle\phi_t^{(n)}(X)\xi,\xi\rangle\geq0. $$ So $\phi^{(n)}$ is positive, and so $\phi$ is completely positive. A similar computation shows that $\phi$ is completely contractive.

Since norm convergence is stronger, and point-norm cluster point will also be ccp. But there is no guarantee that such a cluster point exists. It might in some cases, and it likely won't in most.