TITLE: Self adjoints and unitaries in a banach * algebra
QUESTION [8 upvotes]: Are the spectra of self adjoints and unitaries in banach * algebras necessarily a subset of the reals and the unit circle respectively?  The proofs I know for C* algebras use the continuous functional calculus.

REPLY [8 votes]: As Qiaochu suggests, consider the disc algebra $A$ of continuous functions on the closed unit disc $\overline{D}$ which are analytic in the interior $D$, with supremum norm and the involution $f^*(z) = \overline{f(\overline{z})}$.  In particular, the identity function $z \to z$ is self-adjoint, but its spectrum is $\overline{D}$.