Tracial positive functional
= Tracial positive functional
{title2=$\tau(ab)=\tau(ba)$}
A positive <bounded linear functional> is tracial when its value on a product is unchanged by reversing the two factors. When normalized by $\tau(1)=1$, it is a tracial state. A <vector> functional $\tau(a)=\langle a\Omega,\Omega\rangle$ is faithful if the <vector> is separating, since $\tau(a^*a)=\|a\Omega\|^2$. Its <trace> identity makes the associated <Tomita operator> isometric.