I'm trying to solve some problems on Von Neumann algebra and I got the questions below.
$Q1$. By definition we know a factor is a Von Neumann algebra with trivial center, i.e. a center consisting only of scalar operators. Now suppose $M$ is $\textbf NOT$ a factor. Is it true that $M$ certainly has a "central projection" other than the identity? Actually, this statement doesn't seem obvious to me although it might be trivial.
$Q2$. Assume $M$ is a semifinite and properly infinite factor. Does it contain a proper two-sided ideal? It seems the linear span of all finite projections does the job but I don't know how to verify this claim.
If $p$ and $q$ are two projections in a factor, then either $p$ is subequivalent to $q$ or $q$ is subequivalent to $p$. Does this fact help answer $Q2$?
Any help would be highly appreciated.
The centre of a von Neumann algebra is a von Neumann algebra. A von Neumann algebra is generated (as a Banach space!) by its projections.
A factor cannot have a proper ideal, if we require it to be sot-closed. But the norm-closed ideal generated by the finite projections is a proper C$^*$-ideal. You could try and prove that any linear combination of words on finite projections is at distance $1$ from the identity; this approach works even for a non-factor. Or you could notice that a II$_\infty$ factor is always $M\otimes B(H)$, with $M$ a II$_1$-factor, and take $J=M\otimes K(H)$.