From a formula of Kovarik to the parametrization of idempotents in Banach algebra.

Julien Giol (1)

1 LaBAG - Laboratoire Bordelais d'Analyse et Géométrie

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.

Domaines: Analyse fonctionnelle [math.FA]

Julien Giol. From a formula of Kovarik to the parametrization of idempotents in Banach algebra.. Illinois Journal of Mathematics, 2007, 51 (2), pp.429-444. ⟨hal-00288818⟩