Suppose $G$ is a countable discrete group acting on vN algebra $M$, the action is compact. Can we have a topology on Aut$(M)$ such that $\{\sigma_{g}\in \text{Aut}(M):g \in G\}$ form a compact subset in Aut$(M)$? For example for PMP actions on measure space $(X,\mu)$ there is a notion of weak topology in Aut$(X,\mu)$ that is the smallest topology such that the functions $Aut(X)\ni T\rightarrow \mu(TA\Delta B)$ are continuous, for compact actions the automorphisms generated from group form a compact subset. I am asking this for general von Neumann algebra preserving faithful normal state $\varphi$. What is the weak topology analog in $Aut(M)$?
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2$\begingroup$ What do you mean by a compact action? $\endgroup$– Adrián González PérezCommented May 21, 2019 at 9:09
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$\begingroup$ Compact action means the unitary representation induced from the action is the each element $\xi$ $\in$ $\mathcal{H}$ the set $\{\Pi(g)\xi:g \in G\}$ is compact subset of $\mathcal{H}$. $\endgroup$– user136400Commented May 21, 2019 at 10:05
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4$\begingroup$ If you equip $\mathrm{Aut}(M)$ with the silly topology which only has open sets $\emptyset$ and $\mathrm{Aut}(M)$, then every subset is compact. If you don't want silly answers like that, you need to be more specific. $\endgroup$– Jamie GabeCommented May 21, 2019 at 21:02
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$\begingroup$ Now you can explain Jamie Gabe!! $\endgroup$– user136400Commented May 23, 2019 at 7:08
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