Suppose $T_{n}$ converging $T$ in vN algebra $M$ in weak operator topology, can we conclude $||T_{n}||$ is uniformly bounded? Another question if a linear functional $\varphi$ is continuous in unit ball in $M$ in weak operator topology does it imply that $\varphi$ is normal that means in the predual?

## 1 Answer

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Suppose $M$ acts on a Hilbert space $H$, for a fixed $x\in H$, the set $\{\langle T_nx,y\rangle: n\in\mathbb N\}$ is bounded for all $y\in H$, so $\{T_nx: n\in\mathbb N\}$ is bounded by the principle of uniform boundedness, and again using principle of uniform boundedness we conclude that $\{T_n: n\in\mathbb N\}$ is also bounded.

Yes to the second question, see for example Theorem 1.10 in "lectures on von Neumann algebras" by "S. Strătilă, L. Zsido".