Take $U=\mathbb{D}\_2\setminus \overline{\mathbb{D}\_1}$ ($\mathbb{D}\_r$ is the open disc centered at 0 with radius $r$) and consider the space $A(U)$ of all functions on $\overline{U}$ which are holomorphic in $U$ and admit a continuous extension to $\overline{U}$ (with obvious operations and the supremum norm).


Is $A(U)$ isomorphic to the disc algebra as a Banach algebra?


EDIT: I tried in fact, to find out, if all unital multiplicative linear functional on $A(U)$ are point-evaluations. Are they?



Reversed disc algebra?
No. The spectrum of $A(U)$ is $\overline{U}$, while the spectrum of the disc algebra is $\overline{\mathbb D\_1}$, and these two spaces aren't homeomorphic.


