Let $U$ be the bilateral shift operator on $\ell^2(\mathbb Z)$, and let $T=U+U^*$. How to calculate the spectrum $\sigma(T)$? And how to show there is no cyclic vector for the action of $C^*(T,I)$? Further how to decompose this representation to direct sum of cyclic representations?
Something I know so far: if there is a cyclic vector for the action of $C^*(T,I)$, then the commutant $C^*(T,I)'$ will be commutative. But I don't know how to identify this commutant.