I have this question: Is it true that if $\tau$ is a pure state on a $C^*$-algebra $A$ and $(H_{\tau},\pi)$ is the GNS represenation of $A$ with respect to $\tau$, then is it true that the weak closure of $\pi(A)$ is the whole of $B(H_{\tau})$?
What do I have to show for this, if this is true? I just am confused on how to proceed for the proof. A hint would do.
Thanks for the help!!
If a representation $(H,\pi)$ of $A$ is irreducible, then $\pi(A)'=\mathbb C1$, where $1=\text{id}_{H}$. Thus $\pi(A)''=B(H)$, and the von Neumann bicommutant theorem implies that $\pi(A)$ has weak closure $B(H)$.