Let $K$ be a circle. Describe the spectra of two subalgebras of $C(K)$

Question

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.