Let (l^2), let , and let . Show that
Given (l^2) is a C* -Algebra where for each operator T in B(l^2), T* = is the adjoint of T is that true? or only for B(H)?
The adjoint of the (forwards) unilateral shift is the backwards unilateral shift. So the product is the identity (if you shift forwards and then backwards you get back to where you started). But is not invertible because the backwards shift kills off the first basis vector. Thus but .
In fact, 0 is the only number that can be in the spectrum of st but not in the spectrum of ts (where s, t are elements of a C*-algebra). There is a theorem which says that .