Definition 12.3.18 (Tracial vNA).label Let $\tau(x)\define \inner{\de_e}{x\de_e}$ for $x\in L(G)$ then $\tau(1)=\inner{\de_e}{\de_e}=1$ and $\tau(\lam(s_{1})\lam(s_{2}))=\inner{\de_e}{\de_{s_1s_2}}\neq0\iff s_{1}s_{2}=e$. $\tau$ is continuous w.r.t. the WOT.

A tracial von Neumann algebra is a vNA $M$ equipped with a faithful tracial state that is WOT continuous restricted to the unit ball.