In a finite von Neumann algebra, the unique tracial state serves as one, then for a general von Neumann algebra, does it exist?
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$\begingroup$ in line with Dmitri's answer below: when you say that there is a unique tracial state, I think you mean to say that on a finite factor there is a unique faithful normal tracial state. Otherwise, consider $\ell^\infty$. $\endgroup$– Yemon ChoiCommented Nov 24, 2010 at 17:52
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$\begingroup$ I think you need to reword your question to be a little more precise. Presumably you are really interested in the case of $\sigma$-finite, properly infinite von Neumann algebras? eom.springer.de/v/v096900.htm $\endgroup$– Yemon ChoiCommented Nov 24, 2010 at 17:54
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$\begingroup$ On the topic of precision, I assume you want your expectation to be faithful (otherwise any state would do) and normal. $\endgroup$– Martin ArgeramiCommented Nov 25, 2010 at 13:53
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2 Answers
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Using direct integral decomposition, also known as reduction theory, one can reduce the problem to the case of a factor. A conditional expectation in this case is a state. Every factor admits a state, but only σ-finite factors admit faithful states. Thus if you require the conditional expectation to be faithful, all factors in the direct integral decomposition must be σ-finite, otherwise no additional conditions are needed to ensure the existence of a conditional expectation.
answered Nov 24, 2010 at 15:07
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The answer is yes, provided that $M$ has a faithful normal semifinite weight (this always exists) that is also semifinite when restricted to the centre (this I'm not so sure how easily can happen).
When $M$ has a faithful normal semifinite weight $\varphi$, with $\varphi|_{Z(M)}$ semifinite, consider the modular group $\sigma_t^\varphi$ associated with $\varphi$. For each $t\in\mathbb{R}$, $\sigma_t^\varphi$ is an automorphism of $M$, and in particular it preserves its centre. This means that $$ \sigma_t^\varphi(Z(M))=Z(M), \ \ t\in\mathbb{R} $$
These conditions, by Takesaki's Theorem (IX.4.2 in Takesaki 2, or JFA1972) are equivalent to the existence of a conditional expectation $E:M\to Z(M)$, with $\varphi\circ E=\varphi$. This last condition forces $E$ to be faithful and normal.
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$\begingroup$ You certainly do not need any modular theory to see this. $\endgroup$ Commented Nov 25, 2010 at 15:51
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1$\begingroup$ That wouldn't surprise me, but off the top of my head I wouldn't know how to do it in another way. $\endgroup$ Commented Nov 25, 2010 at 17:41
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2$\begingroup$ I think that the main point of Takesaki's theorem is that it characterizes the subalgebras for which a conditional expectation exists. However, if you just want to have a conditional expectation onto the center then why don't you proceed as Dmitri Pavlov in his answer. Of course there are some technicalities hidden in the direct integral decomposition and making measurable choices etc, but that is something you are facing anyway. I think that Takesaki's theorem is difficult (relies on modular theory) in the factor case; but then a direct integral approach has to be followed anyway. $\endgroup$ Commented Nov 25, 2010 at 20:44
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4$\begingroup$ I agree that Takesaki's theorem is difficult, but it is a proven result and it doesn't use direct integrals (I try to avoid them precisely because of the technicalities). In Dmitri's argument, I'm not sure exactly which vN algebras can be decomposed into a direct integral of $\sigma$-finite factors, and I don't immediately see if the expectation obtained is faithful and normal. $\endgroup$ Commented Nov 25, 2010 at 23:54