I am not getting anywhere with task E.5.10 of Lectures on von Neumann algebras of Stratila and Zsidó. Do you have any tips?
Let $\phi$ be a w-continuous form on $M$ and $\phi=R_{\nu}\,|\phi|$ its polar decomposition. Then the polar decomposition of the form $\phi^*$ is
$\phi^*=R_{\nu^*}\,|\phi^*|$, $\quad|\phi^*|=L_{\nu^*}R_{\nu}\,|\phi|$.
In particular: $\phi=L_{\nu^*}\,|\phi^*|$, $\quad vv^*= \textbf{s}(|\phi^*|)$
Thanks for your help.