Structure Theory of Complemented Banach Algebras

Abstract

If A is an H*-algebra, then the orthogonal complement of a closed right (left) ideal I is a closed right (left) ideal P. Saworotnow (7) considered Banach algebras which are Hilbert spaces and in which the closed right ideals satisfy the complementation property of an H*-algebra. In our right complemented Banach algebras we drop the requirement of the existence of an inner product and only assume that for every closed right ideal I there is a closed right ideal IP which behaves like the orthogonal complement in a Hilbert space (Definition 1). Thus our algebras may be considered as a generalization of Saworotnow's right complemented algebras.