  We study several notions of ultraproducts of von Neumann algebras from a
unifying viewpoint. In particular, we show that for a sigma-finite von Neumann
algebra $M$, the ultraproduct $M^{\omega}$ introduced by Ocneanu is a corner of
the ultraproduct $\prod^{\omega}M$ introduced by Groh and Raynaud. Using this
connection, we show that the ultraproduct action of the modular automorphism
group of a normal faithful state $\varphi$ of $M$ on the Ocneanu ultraproduct
is the modular automorphism group of the ultrapower state
($\sigma_t^{\varphi^{\omega}}=(\sigma_t^{\varphi})^{\omega}$). Applying these
results, we obtain several phenomena of the Ocneanu ultraproduct of type III
factors, which are not present in the tracial ultraproducts. For instance, it
turns out that the ultrapower $M^{\omega}$ of a Type III$_0$ factor is never a
factor. Moreover we settle in the affirmative a recent problem by Ueda about
the connection between the relative commutant of $M$ in $M^{\omega}$ and
Connes' asymptotic centralizer algebra $M_{\omega}$.
