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Let A be a C* algebra of operators on a Hilbert space H. Can it happen that for some x in H the set Ax is dense in H but it is not the whole H?

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The answer is yes, as per Nik Weaver's hint: E.g. $C[0,1]$ acting by multiplication on $L_2[0,1]$.

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    $\begingroup$ Some wisdom I learned from Chuck Akemann: first check the commutative case, then check the $2\times 2$ matrices. If both of those cases work, there's a good chance it's true in general. $\endgroup$
    – Nik Weaver
    Jan 24, 2013 at 3:58

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