8
$\begingroup$

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.

  • 2
    I expect the answer to be no due to lack of spectral permanence but I don't know enough about general Banach algebras to produce a counterexample. I think the disc algebra is a Banach *-algebra with the *-operation given by conjugation of the coefficients in the Taylor series and it seems like a counterexample can be constructed from here, but I haven't checked the details.2012-09-09

1 Answers 1

8

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 $\overline{D}$.