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?
 A: 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.
A: 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. 
