FROM A FORMULA OF KOVARIK TO THE PARAMETRIZATION OF IDEMPOTENTS IN BANACH ALGEBRAS

摘要

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.