Example of a Banach algebra with identity $e$ such that $\|e\| = t$, where $t \geq 1$.

Question

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.

Answer 1

Let $X = (X, \lVert · \rVert)$ be a Banach algebra and $t ≥ 1$. Then $X$ with $t\lVert · \rVert$ is also a Banach algebra.

Thus, for example $ℝ$ with the norm $\lVert · \rVert_t \colon ℝ → [0..∞), x ↦ t|x|$ becomes a Banach algebra with $\lVert 1 \rVert_t = t$. Submultiplicativity of the norm follows from $t ≤ t^2$.