Let $M$ be a von Neumann algebra with faithful normal state $\varphi$, and $G$ be a group action on $M$ preserving $\varphi$. Then is it true \begin{align*} L^{p}(M\rtimes G, \varphi^{M\rtimes G})\cong L^{p}(M,\varphi)\rtimes G? \end{align*}
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$\begingroup$ The right hand side of the isomorphism doesn't make sense to me a priori. I assume the crossed product of the left is the spatial crossed product of von Neumann algebras, but how do you define the crossed product on the right? $L^p$ is not a von Neumann algebra. $\endgroup$– Adrián González PérezCommented Nov 22, 2019 at 9:23
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$\begingroup$ Btw, there is a notion of crossed products for operator systems (arxiv.org/pdf/1803.10759.pdf), but it is recent and is not clear to me if that is what you are using. $\endgroup$– Adrián González PérezCommented Nov 22, 2019 at 9:26
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$\begingroup$ @Adrian right side is about crossed product of the operator systems $\endgroup$– user136400Commented Nov 22, 2019 at 10:06
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3$\begingroup$ Then, I may advise you to explain a little bit the definition you are working with and what have you tried (isn't embedding $\mathbf{C} G$ inside $L^p(M) \rtimes G$ and taking closures enough to disprove your identity?) $\endgroup$– Adrián González PérezCommented Nov 23, 2019 at 11:33
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