## Illinois Journal of Mathematics

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

Julien Giol

#### 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.

#### Article information

Source
Illinois J. Math., Volume 51, Number 2 (2007), 429-444.

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258138422

Digital Object Identifier
doi:10.1215/ijm/1258138422

Mathematical Reviews number (MathSciNet)
MR2342667

Zentralblatt MATH identifier
1159.46026

#### Citation

Giol, Julien. From a formula of Kovarik to the parametrization of idempotents in Banach algebras. Illinois J. Math. 51 (2007), no. 2, 429--444. doi:10.1215/ijm/1258138422. https://projecteuclid.org/euclid.ijm/1258138422