This question could be way below the level of MO, so apologies in advance. I posted the same question in MS about 10 days ago without a definitive answer so far.
Let $A$ be a Banach algebra with the property $\hspace{4mm}q=pq=qp \Rightarrow \|q\|\leq \|p\|\hspace{4mm}$ whenever $p,q\in A$ are idempotents.
Is there a term coined to the algebras with this property in the literature?
For example, $\ell^2$ with pointwise operations has this property, whereas its unitization $\ell^2\oplus\mathbb{C}$ does not.
