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Suppose $K=\{\lambda\in\mathbb{C}: 1<\vert\lambda\vert<2\};$ put $f(\lambda)=\lambda$. Let $A$ be the smallest closed subalgebra of $C(K$) that contains $1$ and $f$. Let $B$ be the smallest closed subalgebra of $C(K$) that contains $f$ and $\frac{1}{f}$. Decribe the spectra $\sigma_{A}(f)$ and $\sigma_{B}(f)$. Do the same when $K$ is a circle. Please help me.

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In order to get the best possible answers, it is helpful if you say what your thoughts on this problem are; this will prevent people from telling you things you already know, and help them give their answers at the right level. See also How to ask a homework question. –  user53153 Jan 1 '13 at 20:02
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