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?

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    $\begingroup$ I suggest you write and accept your answer, so that the question does not remain unanswered. $\endgroup$ – 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.


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