There is the following result:
Suppose $X$ is a normed algebra with identity $e$. Then $\|e\| \geq 1$.
I am looking for an example to show that even a Banach algebra $X$ with identity $e$ not necessarily satisfies $\|e\| = 1$. A general example such that $\|e\| = t$, where $t \geq 1$ would be even more interesting.
I am aware of several examples of Banach algebras with identity, but these all have an identity with norm $1$. As algebras have exactly one identity, I think I cannot use the well-known Banach algebras as an example.
Any help or comment is highly appreciated.