TITLE: Commutative unital Banach algebra not isomorphic to $C(X)$ for some compact Hausdorff space $X$
QUESTION [5 upvotes]: According to some lecture notes I am reading, "it is not so difficult to find an example of a commutative unital Banach algebra which is not isomorphic to $C(X)$ for some compact Hausdorff space $X$"...
Well I was thinking about using the disk algebra $\mathcal{A}$ of continuous functions on the disk $D = \{z \mid |z| \leq 1\}$ which are holomorphic on the interior of $D$, with the sup norm. 
This should be a commutative unital Banach algebra. However, since it is the beginning of the chapter about the Gelfand transformation, I would like to prove that $\mathcal{A}$ is not isomorphic to $C(X)$ without using Gelfand's result.
What other tools could I use to prove this fact?

REPLY [2 votes]: Every $C^*$ algebra of dimension 
at least $2$  must have zero divisor but the disc algebra has no zero divisor.
Proof of existence of zero divisor is based on decomposition $x=x^+ -x^-$ for a self adjoint element $x$. This decomposition is regarded 
as classical decomposition $f=f^+ -f^-$ with $f^+.f^-=0$ for real valued function $f\in C(X)$.