Let $M$ be a von Neumann algebra. In Takesaki's book, the trace $\tau$ on $M$ is defined as follows:
$\tau$ is a function defined on the positive cone $M_{+}$ with values in $[0, +\infty]$ satisfying some conditions.
My question: Can we define $\tau$ on $M$?
When we define a tracial state $\phi$ on a $C^*$–algebra $A$, the domain of $\phi$ is $A$, rather than $A^+$.
Note the second square bracket in $[0,\infty]$. What Takesaki is defining is a tracial weight. Many von Neumann algebras have no tracial states, but do have faithful semifinite tracial weights. These precisely those with no type III central summand, and where the identity is infinite (i.e., they have at least a component of type I$_\infty$ of II$_\infty$).
The typical example is the trace on $B(H)$ with $H$ infinite-dimensional.