From a formula of Kovarik to the parametrization of idempotents in Banach algebras

Abstract

If $p,q$ are idempotents in a Banach algebra $A$ and if $p+q-1$ is invertible, then the Kovarik formula provides an idempotent $k(p,q)$ such that $pA=k(p,q)A$ and $Aq=Ak(p,q)$. We study the existence of such an element in a more general situation. We first show that $p+q-1$ is invertible if and only if $k(p,q)$ and $k(q,p)$ both exist. Then we deduce a local parametrization of the set of idempotents from this equivalence. Finally, we consider a polynomial parametrization first introduced by Holmes and we answer a question raised at the end of his paper.