Let denote the matrix units in . Show that is a closed ideal in (where is identified with ). Hence (unless , in which case show that ). Deduce that for all i,j, and so .
Suppose we have a simple C*-algebra, say . Show that is also simple.
My first guess would be to try and obtain some contradiction by assuming has some closed ideal other than or and then find that will then have some closed ideal other than or which will conclude the proof.
However, I'm having some trouble with the particulars...
We can suppose that there is some sequence that converges to . Then we have and also for every .
How can we now use this to construct a closed ideal in ? (If it is at all possible)