# Does a conditional expectation from a von Neumann algebra to its center exist?

In a finite von Neumann algebra, the unique tracial state serves as one, then for a general von Neumann algebra, does it exist?

• 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 $\ell^\infty$. – Yemon Choi Nov 24 '10 at 17:52
• 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 – Yemon Choi Nov 24 '10 at 17:54
• On the topic of precision, I assume you want your expectation to be faithful (otherwise any state would do) and normal. – Martin Argerami Nov 25 '10 at 13:53

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.
• 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. – Martin Argerami Nov 25 '10 at 23:54