Let $M$ be a $\mathrm{II}_{1}$ factor. Does there exist a conditional expectation from $M^{\otimes 2}$ to $M$ preserving the trace $\tau^{\otimes 2}$?
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1$\begingroup$ Do you see $M$ as one tensor component of $M \otimes M$? If an inclusion of factors preserves the traces it extends to the preduals. Dualizing you get an expectation. It seems that $M \otimes \mathbb{C} 1 \subset M \otimes M$ is trace-preseving. $\endgroup$– Adrián González PérezCommented Nov 12, 2019 at 11:41
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$\begingroup$ there are many embeddings of $M$ to $M\otimes M$, are you talking about $x\rightarrow x\otimes I$? $\endgroup$– user136400Commented Nov 12, 2019 at 12:05
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1$\begingroup$ Yes, but that is precisely what should be clarified in the question. $\endgroup$– Adrián González PérezCommented Nov 12, 2019 at 13:24
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1 Answer
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Yes. If $N$ is any subalgebra of a II$_1$ factor, then there exists a trace preserving conditional expectation from the ambient II$_1$ factor onto $N$. The proof can be found in any standard text book on von Neumann algebras.
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$\begingroup$ If anyone wants a more specific reference that's open to the public, there's this article by Takesaki. Specifically, apply the main theorem using the trace as the semifinite normal weight, and observe that the modular automorphism group is trivial for a trace, so the invariance condition is automatic. The proof can be done more simply in this case, and makes a nice exercise in understanding the general case. $\endgroup$ Commented May 11, 2020 at 16:55