I needed some insight into spectra in Banach algebras. To this end I tried to calculate a few examples. I will post them not all in one post. Could somebody please tell me if they are correct? Here goes:
Example 3: Let $C$ denote the algebra of continuous functions $f:S^1 \to \mathbb C$ endowed with the $\sup$-norm and let $p(z) = z$. Then $p(z)-\lambda$ is not invertible if and only if $\lambda \in S^1$. Hence $\sigma_C(p) = S^1$.
Example 4: Let $D$ denote the closure of the algebra generated by $1$ and $p(z)=z$ defined on $S^1$ endowed with the $\sup$-norm. Then $p(z)-\lambda = z-\lambda$ is not invertible for any $\lambda$. Hence $\sigma_D (p) = \mathbb C$.
Example 3 is correct, for $\lambda \notin S^1$, $z \mapsto \frac{1}{z-\lambda}$ is a continuous function on the unit circle, and hence $p(z) - \lambda$ is invertible. On the other hand, for $\lambda\in S^1$, $p(z)-\lambda$ has a zero on $S^1$ and is therefore not invertible, thus $\sigma_C(p) = S^1$.
Example 4, however, is not correct. The spectrum of an element in any Banach algebra is compact, therefore $\sigma_D(p) = \mathbb{C}$ is impossible. Since
$$\max \{\lvert\lambda\rvert : \lambda \in \sigma(a)\} \leqslant \lVert a\rVert,$$
we have $\sigma_D(p) \subset \overline{\mathbb{D}}$. In fact, since $D$ is (isometrically isomorphic to) the disk algebra, the continuous functions on $\overline{\mathbb{D}}$ that are holomorphic in $\mathbb{D}$, we see that $\sigma_D(p) = \overline{\mathbb{D}}$.