Let be a C*-algebra and let be a rank-one projection in , the space of compact operators on some Hilbert space .
I would like to construct a projection in the multiplier algebra such that
here is my current idea:
consider the mapping such that . Here
It can easily be shown that , however I'm having some trouble in showing that . Is it possible?
Now I see that my original idea was wrong. For the mapping
it is easy to show that , but once again I am missing something in determining ?
The way I am thinking about this is to have some kind of inner product in order to determine .
Another thing is if this is in fact the projection I am looking for then
now if we let then
but is ?
Looking at the problem again, I don't see why you need the map at all. Simply define , where 1 is the identity element in . The point is that if is a rank-one projection then is always a scalar multiple of . In fact, if is a unit vector in the range of then , where is the state on defined by