The structure of the set of idempotents in a Banach algebra

Abstract

We study here the algebraic, geometric, and analytic structure of the set of idempotent elements in a real or complex Banach algebra. A neighborhood of each idempotent in the set of idempotents forms the set of idempotents in a Rees product subsemigroup of the Banach algebra. Each nontrivial connected component of the set of idempotents is shown to be a generalized saddle, a type of analytic manifold. Each component is also shown to be the quotient of a (possibly infinite dimensional) Lie group by a Lie subgroup.