ideals, projections and factors in VN algebras

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 $J=M\otimes K(H)$.