I was going through the proof of the Gelfand Naimark Theorem for the Unital Commutative Banach Algebras.
In proving that each character has norm equal to $1$, we used the fact that $\|e\| = 1$ where $e$ is the identity of the Banach algebra. However, since not every Banach algebra satisfies $\|e\| = 1$, is this result not true for those Banach algebras? Or is there a way around to prove the result in that case too?
You are right, my previous attempt was wrong:
here are examples that answer your question: take $A=C(X)$ to be the Banach algebra of all continuous function on some compact space (say $X=[0,1]$, or take $X$ to be a point so that $A=\mathbb{C}$) and define a new norm on $A$ by $\|f\|_r:= r \sup_{x\in X} |f(x)|$, where $r>1$ is some (fixed) number. Then $A$ is still a Banach algebra with this new norm. Take a character of the form $\chi(f):=f(x_0)$ for some $x_0\in X$. This character has norm $1/r$, which can be any number in $(0,1)$.
