On complemented and annihilator algebras

The purpose of this paper is twofold. In [6] Tomiuk gives a representation theorem for a topologically simple right complemented algebra that is also an annihilator algebra. We strengthen this and then give a converse, so as to characterise right complemented algebras among respectively primitive Banach algebras and primitive annihilator Banach algebras. Our second aim is to investigate the relationship between the different annihilator conditions—left annihilator, right annihilator, annihilator, and dual—when imposed on a complemented algebra. Tomiuk [6] has already shown that a right complemented semisimple algebra that is a left annihilator algebra is an annihilator algebra; further, a topologically simple bi-complemented algebra that is also an annihilator algebra is dual. We show that for a topologically simple right complemented algebra all four annihilator conditions are equivalent. Further, for a semi-simple Banach algebra the first three are equivalent provided it is right complemented, and if it is also left complemented, then they are equivalent to duality.