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Let $A$ be a Banach algebra with the property $\big(q=pq=qp \Rightarrow \|q\|\leq \|p\|\big)$ whenever $p,q\in A$ are idempotents.

Is there a term coined to the algebras with this property in the literature?

For an example, $\ell^2$ with pointwise addition and multiplication has this property, whereas its unitization $\ell^2\oplus\mathbb{C}$ does not.

Onur Oktay
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  • Since general Banach algebras are unlikely to contain too many projections, any condition based only on projections is bound not to be too useful as it will often hold vacuously. – Ruy Jul 27 '22 at 11:57

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