Every type II$_1$ von Neumann algebra has a unique faithful trace

Question

Here I am working with the definition that a von Neumann factor $M$ is a type-II factor if there is no non-zero, minimal projection but there are non-zero finite projections. It is a type-II$_1$ factor if it is a type-II factor and the identity is finite.

I've read that every type-II$_1$ factor $M$ has a unique, faithful trace. Namely there is a unique linear map $\omega: M \to [0 , \infty]$ so that:

1. It is a trace, i.e. $\omega \left( x y \right) = \omega \left( y x \right)$
2. It is faithful $\omega \left( x^{*}x \right)=0$ implies $x=0$.

However I haven't been able to find a proof of this statement anywhere. It would be appreciated if someone could provide a proof, reference or suggest how to approach the proof myself.

Answer 1

This is done in several places, for instance in chapter 8 in Kadison-Ringrose; it's done in more generality, as they show that any finite von Neumann algebras has a unique central valued trace.

The "Murray-von Neumann way" is done in Sunder's An Invitation to von Neumann Algebras. Here is the (very rough) idea.

• Show that two projections with the same central carrier are comparable. Conclude that in a factor all projections are comparable.

• Prove an "Euclidean Algorithm" for projections: given projections $p,q\in M$, there exist projections $q_1,\ldots,q_m,r\in M$ such that $q_j\sim q$, $r\prec q$, and $p=r+\sum_jq_j$.

• If $q|1$, that is if there exist $q_1,\ldots,q_m$ with $q_j\sim q$ and $\sum_jq_j=1$, define $\tau(q)=1/m$.

• If $q$ admits a subprojection $q_1$ that divides both $1$ and $q$, define $\tau(q)=n/m$ where $m$ is the number of copies of $q_0$ in $1$, and $n$ the number of copies of $q_0$ in $q$.

• For an arbitrary projection $p$, if $p=\sum_jq_j$ with each $q_j$ "rational" as above, define $\tau(p)=\sum_j\tau(q_j)$.

• Note that the above decomposition of $p$ can always be done, as in a II$_1$ factor we have infinite divisibility of projections. With a bit of care, one can show that you can always "halve" a projction $p$, in the sense that there exist subprojections $p_1,p_2$, with $p_1p_2=0$, $p_1\sim p_2$, and $p_1+p_2=p$.

• Now extend $\tau$ to linear combinations of pairwise orthogonal projections, and by taking limits to selfadjoint operators. And finally, by linearity again, to arbitrary elements.