Complemented Banach*-algebras.

Abstract

General complemented Banach algebras have been studied by B.J. Tomiuk [Canadian J. of Math. Vol. 14 (1962), 651--659], and recently he and F.E. Alexander have contributed to the study of complemented B*-algebras [Trans. AMS 1969]. In this thesis we extend their work to complemented Banach*-algebras and show their relation to annihilator and dual Banach*-algebras. If A is a semi-simple complemented Banach*-algebra in which x*x = 0 implies x = 0, then A is an A*-algebra which is a dense subalgebra of a dual B*-algebra M; M is uniquely determined up to *-isomorphism. We give several characterizations of duality for A*-algebras, some of which are expressed in terms of complementors. We show, in particular that if A is a dense 2-sided ideal of a B*-algebra, then A is dual if and only if it is complemented. Every complemented completely continuous A*-algebra is dual. Let A be an A*-algebra contained in a B*-algebra M, and let p, q be complementors on M and A respectively. Using the properties of continuous complementors on B*-algebras, we obtain conditions on M, A and the complementors p and q such that: (a) The mapping I → cl (I)p ∩ A on the closed right ideals I of A is a complementor on A (called the complementor on A induced by p). (b) The mapping R → cl(R ∩ A)q) on the closed right ideals R of M is a complementor on M (called the complementor on M induced by q). Finally we discuss an example of a complemented A*-algebra.