Meeting 5: Zelin Yi

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A state of a C*-algebra is a positive linear functional of norm one. Let $\varphi$ be a state on a von Neumann algebra $M \subseteq \mathbb{B}(H)$. We say that $\varphi$ is normal if $\varphi(\bigvee_{\alpha} T_\alpha)=\lim \varphi(T_\alpha)$ for every bounded, increasing net $(T_\alpha)$ in $M$. We say that $\varphi$ is completely additive if $\varphi(\bigvee_\alpha p_\alpha)=\sum_\alpha \varphi(p_\alpha)$ for every collection $(p_\alpha)$ of mutually orthogonal projections in $M$. In this meeting, we established the following equivalences.

Theorem 5.1: For a state $\varphi$ of a von Neumann algebra $M \subseteq \mathbb{B}(H)$, the following are equivalent:

1. $\varphi$ is normal
2. $\varphi$ is completely additiive.
3. $\varphi$ is $\sigma$-weakly continuous.
4. $\varphi$ is $\sigma$-strongly continuous.
5. There exists $v_1,v_2,\ldots \in H$ with $\sum \|v_i\|^2 =1 $ such that $\varphi$ is given by $T \mapsto \sum \langle T v_i,v_i \rangle$. In other words, $\varphi$ is a vector state, after possibly inflating the Hilbert space.

NOTES NOT COMPLETE!!

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