Let $(M, \tau)$ be a $\mathrm{II_{1}}$ factor equipped with the faithful normal tracial state $\tau$ acts on $L^{2}(M,\tau)$ is in standard form. The question is: if we consider vN algebras $PMP(P\in M)$ acting on $PL^{2}(M,\tau)$ and $QM(Q \in M')$ acting on $QL^{2}(M,\tau)$, are these vN algebra in standard form or not?
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1$\begingroup$ Could you write your last sentence more carefully (don't mind splitting it into two sentences)? $\endgroup$– YCorCommented Aug 12, 2019 at 10:32
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1$\begingroup$ What is your definition of a "standard form"? What have you tried to do? For example, for me a "standard form" involves a "modular conjugation" $J$. What is the modular conjugation on $PL^2$ or $QL^2$? $\endgroup$– Matthew DawsCommented Aug 12, 2019 at 12:54
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1$\begingroup$ OK, if you say you don't know whether PJP will work, write down the conditions that PJP would have to satisfy in order for it to be a modular conjugation, and then tells us where you get stuck $\endgroup$– Yemon ChoiCommented Aug 13, 2019 at 5:20
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1$\begingroup$ @Dimitri obviously P and Q are projections that's why they are said to be corner. $\endgroup$– user136400Commented Aug 13, 2019 at 7:54
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5$\begingroup$ I'm voting to close this question since the OP seems not to have engaged with the hints in the comments, and has been asking a series of small questions without signs of progressing to working things out themselves $\endgroup$– Yemon ChoiCommented Sep 24, 2019 at 12:07
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