Let $\displaystyle \cal{A} = \cal{B}$(l^2), let $\displaystyle a = the \;unilateral \;shift \;on \;l^2$, and let $\displaystyle b=a^* $. Show that $\displaystyle \sigma(ab) \neq \sigma(ba) $

Given $\displaystyle \cal{A} = \cal{B}$(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)?