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?



Cyclic vectors for C* algebras
The answer is yes, as per Nik Weaver's hint: E.g. $C[0,1]$ acting by multiplication on $L\_2[0,1]$. 


