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.
Self adjoints and unitaries in a banach * algebra
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analysis
functional-analysis
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2I 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
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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 $\overline{D}$.