Definition 43.4.6 (Normal Positive Map).label Let $M, N$ be von Neumann algebras and $\phi: M \to N$ be a positive linear map, then the following are equivalent:
- (1)
For any bounded, upward-directed family $A \subset M$, $\phi(\sup(A)) = \sup(\phi(A))$.
- (2)
For any normal, positive linear functional $\omega \in N^{*}$, $\omega \circ \phi$ is a normal, positive linear functional.
- (3)
$\phi$ restricted to bounded subsets of $M$ is continuous with respect to the weak operator topologies.
If $\phi$ is in addition a *-homomorphism, then the above are also equivalent to
- (4)
$\phi$ restricted to bounded subsets of $M$ is continuous with respect to the strong operator topologies.
If the above holds, then $\phi$ is normal.
Proof, [Proposition 2.5.8, AP]. (2) $\Rightarrow$ (3): Let $\omega \in N^{*}$ be a weak-operator continuous functional. By Lemma 43.4.1, there exist positive weak-operator continuous functionals $\bracsn{\omega_k}_{0}^{3}$ such that $\omega = \sum_{k = 0}^{3} i^{k} \omega_{k}$. By assumption, $\omega_{k} \circ \phi$ is weak-operator continuous on bounded subsets of $M$. In which case, $\omega \circ \phi = \sum_{k = 0}^{3} i^{k} \omega_{k} \circ \phi$ is weak-operator continuous on bounded subsets of $M$. Therefore $\phi$ is weak-operator continuous on bounded subsets of $M$.
(3) $\Rightarrow$ (1), (4) $\Rightarrow$ (1): By (1) of Theorem 43.3.1.
(3) $\Rightarrow$ (4)[1]: Suppose that $M$ acts on the Hilbert space $H$, and $N$ acts on the Hilbert space $K$. Let $\angles{x_\alpha}_{\alpha \in A}\subset M$ be a bounded net converging to $x \in M$ in the strong operator topology, then for each $\xi \in H$,
By polarisation, $\wotlim_{\alpha \in A}(x - x_{\alpha})^{*}(x - x_{\alpha}) = 0$. As $\angles{x_\alpha}_{\alpha \in A}$ is bounded,
as well. In particular, for any $\xi \in K$,
Therefore $\phi$ restricted to bounded subsets of $M$ is strong-operator continuous.$\square$
- Central here is the observation that $x_{\alpha} \to 0$ in the strong operator topology i.f.f. $x_{\alpha}^{*}x_{\alpha} \to 0$ in the weak operator topology. keyboard_return
Post a Comment