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 1: Let $A$ be the disk algebra of functions $f: \mathbb D \to \mathbb C$ such that $f$ is holomorphic on the interior of the disk and continuous on the boundary endowed with the $\sup$-norm. Let $p(z) = z$. Then the function $p(z) -\lambda$ is not invertible if and only if there is $z\in \mathbb D$ such that $p(z) - \lambda = 0$ which is the case if and only if $\lambda \in \mathbb D$. That is, the spectrum is $\sigma_A(p) = \mathbb D$ the unit disk around $0$.
Example 2: Let $B$ denote the Banach algebra of continuous functions $f: \mathbb D \to \mathbb C$ endowed with the $\sup$-norm and $p(z) = z$. Then $p(z) -\lambda$ is not invertible if and only if $\lambda \in \mathbb D$. Hence $\sigma_B(p) = \mathbb D$.
The conclusion I reach is that the element $p(z) = z$ has the same spectrum in both the disk algebra and the algebra of continuous functions on the disk. Therefore, it is not clear to me how analyticity influences the properties of the algebra.
(1) Are my example calculations correct?
(2) What desirable property does the disk algebra have that the algebra of continuous function does not have?