35.5 The Disk Algebra
Definition 35.5.1 (Disk Algebra).label Let $D = B_{\complex}(0, 1)$, then the disk algebra $A(D) = H(D; \complex) \cap C(\ol D; \complex)$ is the space of holomorphic functions on $D$ that extend to $\ol D$, equipped with the uniform norm.
Proposition 35.5.2.label Let $f \in A(D)$, then $\sigma(f) = f(\ol D)$.
Proposition 35.5.3.label Let $D = B_{\complex}(0, 1)$ and $A(D)$ be the disk algebra, then the mapping
is a homeomorphism.
Proof. Let $\phi \in \Omega(D)$. By Proposition 33.7.2, $\norm{\phi}_{A(D)^*}= 1$. Let $p$ be the identity polynomial, and $z_{0} = \phi(p)$, then for every $q \in \complex[x]$, $\phi(q) = q(x_{0})$. By density of polynomials in $A(D)$, $\phi(f) = f(z_{0})$ for all $f \in A(D)$. Therefore $\ol D$ is in bijection with $\Omega(D)$.
Since $A(D) \subset C(\ol D)$, $E$ is continuous. As $\ol D$ is compact and $\Omega(D)$ is Hausdorff, $E$ is a homeomorphism by Proposition 5.16.5.$\square$
Post a Comment