Involution on the Disc Algebra

Question

We know that the disc algebra $A(D)$={$f$ in $C(DUBd(D))$: f is analytic on $D$} wher $D$ is the open unit disc in the complex plane is a Banach algebra under the usual sup norm.Is it possible to define an involution on $A(D)$ such that $A(D)$ becomes a C$^*$-algebra? I think it is not possible, but not really able to prove it.Thanks for any help.

Answer 1

A commutative $C^{\ast}$ algebra must be isomorphic to $C(X)$, where $X$ is its maximal ideal space. The maximal ideal space of the disk algebra is $\overline{D}$, and $A$ is not isomorphic to $C(\overline{D})$.

Thus, there is no norm on $A(D)$ which makes it a $C^{\ast}$-algebra.