## Exercises on von Neumann Algebra


\begin{equation*} \lvert uv\rvert\leq \lVert u\rVert\lvert v\rvert. \end{equation*}

Indded,Let $v=w\lvert u\rvert$ and $uv=w'\lvert uv\rvert$ be the Polar Decomposition, then $\lvert uv\rvert^2=w'^*uw\lvert v\rvert$, thus \begin{align*} \lvert uv\rvert^2 & =\lvert v\rvert w^* u^* w' w'^*uw\lvert v\rvert\\ &\leq \lVert w'^*uw\rVert^2\lvert v\rvert^2\leq\l{u}^2\lvert v\rvert^2, \end{align*} Hence $\lvert uv\rvert\leq \lVert u\rVert\lvert v\rvert.$

$\textbf{Theorem}.$ Let $A$ be a $\vN$ on Hilbert space containing $\operatorname{id}_H$, then for any $\sigma$-continuous linear $\tau$ on $A$, there exists $u\in L^1(H)$ such that $$\tau(v)=\operatorname{tr}(uv)$$ for all $v\in A$.
