On the converse of a theorem of Harte and Mbekhta: Erratum to "On generalized inverses in C*-algebras"
We prove that the converse of Theorem 9 in "On generalized inverses in C � -algebras" by Harte and Mbekhta (Studia Math. 103 (1992)) is indeed true. In (3), Harte and Mbekhta give the following theorem (A is a C � -algebra): Theorem 1. A normalized commuting inverse is unique. If a 2 A has a commuting generalized inverse then it is decomposably regular, and A = aA + a 1 (0) with aA \ a 1 (0) = {0}, (1)