Definition 12.3.17 (States and Traces).label Let $\p$ be a linear functional $\p$ on a von Neumann algebra $M$ then $\p$ is

1. (1)

   positive if $\p(x^{*}x)\geq 0$ for all $x\in M$

2. (2)

   faithful if $\p(x^{*}x)=0\implies x=0$

3. (3)

   state if $\p(1)=1$

A positive linear functional $\p$ is a trace if $\p(xy)=\p(yx)$ for all $x,y\in M$. If $\p$ is also a trace then we call it a tracial state

For a vNA $M$ acting on $\cali{H}$, $\xi\in\cali{H}$ is cyclic for $M$ if $M\xi$ is dense in $\cali{H}$. $\xi\in\cali{H}$ is separating for $M$ if $x\xi\neq 0$ whenever $x\neq 0$.

We can observe that in $L(G)$, $\de_{e}$ is cyclic and separating^[1].