# When is the image of a GNS representation WOT-dense?

Given a $C^*$-algebra $A$ and a state $\rho$ on $A$, let $\pi_\rho$ be the corresponding GNS representation on the Hilbert space $H_\rho$. I would like know when the image of $\pi_\rho$ is WOT-dense in $B(H_\rho)$.

Is it sufficient to assume that $\rho$ is a pure state?

• I suggest you write and accept your answer, so that the question does not remain unanswered. – Martin Argerami Apr 21 '13 at 20:24

Ah yes, this is indeed true. $\pi_\rho$ is irreducible if and only if $\rho$ is pure. And a representation is irreducible if and only if its range is SOT-dense, which by the Bicommutant theorem is equivalent to WOT-density.