I have problems with a proof. Let $ mathcal {A} = L _ { infty} ( mu) $ either von Neumann's algebra of essentially related measurable functions. ($ mu $ finished) I have a sequence of continuous maps $$ V_m:[0,T] rightarrow mathcal {A}. $$ I want to show that this sequence converges weakly to a $ V:[0,T] rightarrow mathcal {A}. $ To do this, I show that the images $ bigcup_m V_m ([0,T]$ are weakly relatively compact.

I know that I have foretold $ mathcal {A} _ {*} = L_1 ( mu) $ of $ mathcal {A} $ this $ mathcal {A} subset mathcal {A} _ {*}. $ So I can consider my $ V_m $ as mappings in the prediction $ mathcal {A} _ {*} $. That's what I was advised to do.

1) What I do not understand, that's why should I make the change to work in the predelict $ mathcal {A} _ {*} $ instead of $ mathcal {A} $? I need uniform integrability, but why can not I consider the images $ V_m ([0,T]$ as subsets of $ L_1 ([0,T], mathcal {A}) $ instead of $ L_1 ([0,T], mathcal {A} _ {*}) $?

2) In addition, I do not understand why the low limit of $ V_m $ would need to be continuous, which is apparently, as I was advised to show …

Thank you so much ðŸ™‚