Why is every factor a tensor product of a $\sigma$-finite factor and a factor of type I?

Question

Here I am not assuming the factor is represented on a separable Hilbert space. This is quoted on page 370 of Takesaki II, then a bit later on page 381, and I haven't been able to find a proof prior to this point in the book or in Takesaki I.

Answer 1

It is because a von Neumann algebra is $\sigma$-finite if it has a faithful normal state, there is a partion of unity $1 = \sum_{i\in I} p_i$ by mutually orthogonal projections equivalent to any given projection $p$ in an infinite factor, and such a decomposition induces the isomorphism $M \sim pMp \bar{\otimes} B(\ell^2I)$.

Answer 2

Note that "$\sigma$-finite" is a tricky notion. For example, any ${\rm\ II}_1$ factor, even non-separable, is $\sigma$-finite, because the finite trace is a faithful state.

In any case, the argument one needs is that of Makoto Yamashita, with a few clarifications. Note that for type I and type II factors, the assertion is trivial: for type I, the $\sigma$-finite factor can be taken to be $\mathbb{C}$. For type ${\rm\ II}_1$, it is always $\sigma$-finite. And any type ${\rm\ II}_\infty$ is the tensor of a ${\rm\ II}_1$ with a ${\rm\ I}_\infty$, so again the assertion holds.

This leaves us then with a factor of type III. The question is why does there exist a projection $p$ with $pMp$ $\sigma$-finite. It is well-known that any von Neumann algebra has a faithful semifinite normal weight. From this one can deduce that there has to exist a projection $p$ where the weight is finite. And then one deduces that, restricted to $pMp$, the weight is a faithful normal state. As $M$ is type III, one can construct a family of pairwise orthogonal projections $\{p_j\}_{j\in J}$ with $p_j$ equivalent to $p$ for all $j$. This equivalences can be used to construct a system of "matrix units", from where the isomorphism $$ M\simeq(pMp)\otimes B(\ell^2(J)) $$ follows.

Answer 3

could you tell me which page it is?