Idempotents with arbitrarily large norms in Banach algebras

Question

While nonzero projections in $C^*$-algebras have norm 1, there is no such restriction for idempotents in Banach algebras. What is an example of a Banach algebra that has idempotents of arbitrarily large norm? I am hoping for an example that involves a fairly "natural" Banach algebra.

Answer 1

It's self-adjoint projections in a $C^*$ algebra that must have norm $1$! Which observation actually leads to the example you want: $$\left[\begin{matrix}1&a\\0&0\end{matrix}\right].$$

So there you are, the algebra of $2\times 2$ matrices.

For a commutative Banach algebra with identity and large idempotents: Say $M_n$ is the span of $\left[\begin{matrix}1&n\\0&0\end{matrix}\right]$ and a $2\times 2$ identity matrix. Let $A$ be the set of all sequences $x=(x_1,\dots)$ with $x_n\in M_n$ and $||x_n||$ bounded; take pointwise multiplication and $||x||=\sup||x_n||$. (This is "natural" because it's an algebra of operators on $\ell^2$...)

Answer 2

There are also natural commutative examples.

Take $\ell_p(\mathbb N$) with pointwise multiplication. Consider a sequence $q$ that has $n$ ones and all other entries are zero. Clearly, $qq=q$ and $\|q\| = n^{1/p}$.