Dependence of the Spectrum on the Algebra

If $K$ is a compact subset of $\mathbb {C}$, let $P(K)$ be the closure of the polynomials in $C(K)$. Show that the identity map on polynomials extends to an isometric isomorphism of $P(K)$ onto $P(\hat{K})$.

If $K$ is a compact subset of $\mathbb {C}$, let $R(K)$ be the closure in $C(K)$ of all rational functions with poles off $K$. If $f \in R(K)$, show that $\sigma_{R(K)}(f) = f(K)$. If $f \in P(K)$, show that $\sigma_{P(K)}(f) = \hat{f}(\hat{K})$, where $\hat{f}$ is a natural extension of $f$ to $\hat{K}$.

Let $\mathcal{A}$ and $\mathcal{B}$ be Banach algebras with a common identity such that $\mathcal{B} \subseteq \mathcal{A}$ and $a \in \mathcal{B}$; recall that then $\sigma_{\mathcal{A}}(a) \subseteq \sigma_{\mathcal{B}}(a)$ and $\sigma_{\mathcal{A}}(a)$ and $\sigma_{\mathcal{B}}(a)$ have the same polynomially convex hull. If $\sigma_{\mathcal{A}}(a) \subseteq \mathbb {R}$, show that $\sigma_{\mathcal{B}}(a) = \sigma_{\mathcal{A}}(a)$.

Let $\mathcal{A}$ be a Banach algebra with identity and let $a \in \mathcal{A}$. If $G_1, G_2, \ldots$ are the holes of $\sigma_{\mathcal{A}}(a)$ and $1 \le n_1 \le n_2 \le \cdots$, show that there is a subalgebra $\mathcal{B}$ of $\mathcal{A}$ such that $a \in \mathcal{B}$ and $\sigma_{\mathcal{B}}(a) = \sigma_{\mathcal{A}}(a) \cup \bigcup_{k=1}^\infty G_{n_k}$.

If $\mathcal{A}, \mathcal{B}$, and $a$ are as in Exercise 3 (with $\mathcal{A}$ not abelian and $\mathcal{B}$ a maximal abelian subalgebra of $\mathcal{A}$), show that $\sigma_{\mathcal{A}}(a) = \sigma_{\mathcal{B}}(a)$.

If $K$ is a nonempty compact subset of $\mathbb {C}$ that is polynomially convex, show that the components of $\text{int}\, K$ are simply connected.