Illinois Journal of Mathematics

The structure of the set of idempotents in a Banach algebra

J. P. Holmes

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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.

Article information

Illinois J. Math., Volume 36, Issue 1 (1992), 102-115.

First available in Project Euclid: 19 October 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46H20: Structure, classification of topological algebras
Secondary: 22E65: Infinite-dimensional Lie groups and their Lie algebras: general properties [See also 17B65, 58B25, 58H05]


Holmes, J. P. The structure of the set of idempotents in a Banach algebra. Illinois J. Math. 36 (1992), no. 1, 102--115. doi:10.1215/ijm/1255987609.

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