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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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Did you check the commutative case? –  Nik Weaver Jan 23 '13 at 23:07
    
Thank you. That answers my question. –  Nemo Jan 24 '13 at 2:54
    
You might be interested in the article "Somewhere dense orbits are everywhere dense" by Bourdon and Feldman, home.wlu.edu/~feldmann/Papers/SomewhereDense.html –  Jochen Wengenroth Jan 29 '13 at 7:39
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up vote 2 down vote accepted

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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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. –  Nik Weaver Jan 24 '13 at 3:58
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