Exercise 21, § 6

Let S be the closed unit disk in $\mathbf{C}$. Let A be the involutive Banach algebra of continuous functions from S into $\mathbf{C}$ which are holomorphic in the open unit disk, where the involution is given by $f^*(z) =\overline{f(\overline{z})}$. There exists a hermitian element of A whose spectrum is not contained in $\mathbf{R}$.