If $A$ be a unital abelian Banach algebra and contains an idempotent $e$ (that is $e=e^{2}$) other than $0$ and $1$ , then help me to show that $\Omega(A)$ is disconnected.
$\Omega(A)$ is the set of characters on $A$, that is, the set of non-zero homomorphisms from $A$ to $\Bbb C$.
On
Let $x_0\in A\setminus\{\mathbf 1,0\}$ such that $x_0^2=x_0$.
Let $S_1:=\{h\in \Omega(A),h(x_0)=0\}$ and $S_2:=\{h\in \Omega(A),h(x_0)=1\}$. This form a partition of $\Omega(A)$. As these sets are closed, what we have to show is that $S_1$ is not empty.
If it was, then by this thread, the spectrum of $x_0$ would be reduced to $0$. The formula of spectral radius gives a contradiction.
Let $0\neq e\neq 1$ be an idempotent in $A$. Then prove that $\hat{e}:\Omega(A)\rightarrow \{0,1\}$ defined by $\phi\mapsto \phi(e)$ is continuous and onto.