******************************
If x is a unit vector in a Hilbert space H with dimension at least 2,
then D = {T Î B(H); Tx = 0} is a closed left ideal in the C*-algebra B(H). If D has a left
approximate identity {Sa} and h ¹ 0 is a vector in H such that < x,h > = 0, then x[`(Ä)]h Î D and so lima Sa(x[`(Ä)]h) = x[`(Ä)]h.
Thus lima||(Sax-x)[`(Ä)]h|| = lima||Sax-x||||h|| = 0, hence 0 = lima||Sax-x|| = ||x||, a contradiction. Thus D has no left approximate
identity.
Note that for z1 and z2 in H the rank one operator z1[`(Ä)] z2 is defined by
|