Generalized inverses in C*-algebras, 2

We first complete, and second extend, the work begun in the paper "Generalized inverses in C*-algebras". The paper, which recaptured the elementary theory of generalized inversesin the context of C*-algebras, contained an omission: it left open whether or not generalized inverses actually exist. Here, first, we give a condition (Theorem 4) for the existence of a generalized inverse of an element of a C*-algebra. Second, the paper contained a "right-handed" inequality to do with minimizing ||ax-c|| (and a corresponding "left-handed" one to do with minimizing ||xb-c||). Here, we obtain a "double-sided" result on minimizing ||axb-c|| (which subsumes the results of the paper on minimizing ||ax-c|| and ||xb-c||).