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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6$\begingroup$ Did you check the commutative case? $\endgroup$– Nik WeaverCommented Jan 23, 2013 at 23:07
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$\begingroup$ You might be interested in the article "Somewhere dense orbits are everywhere dense" by Bourdon and Feldman, home.wlu.edu/~feldmann/Papers/SomewhereDense.html $\endgroup$– Jochen WengenrothCommented Jan 29, 2013 at 7:39
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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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10$\begingroup$ 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. $\endgroup$ Commented Jan 24, 2013 at 3:58