Generalized inverses in C*-algebras

In this paper we present the elementary theory of generalized inverses in the context of C*-algebras. We recapture and, where necessary, modify some of the well-known results of [6], [7]. The methods are, of course, algebraic rather than geometric: there can be no references to ranges or kernels. For this reason we are unable to demonstrate the existence of the Moore-Penrose inverse (and hence the existence of generalized inverses): although we are able to give a criterion, (Theorem 3), a variant of Penrose's [6, Theorem 1] for there to be no. more than one generalized inverse. Other results are recaptured in this setting.