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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm184-2-4, author = {Xavier Mary}, title = {On the converse of a theorem of Harte and Mbekhta: Erratum to "On generalized inverses in C*-algebras"}, journal = {Studia Mathematica}, volume = {187}, year = {2008}, pages = {149-151}, zbl = {1142.46025}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm184-2-4} }
Xavier Mary. On the converse of a theorem of Harte and Mbekhta: Erratum to "On generalized inverses in C*-algebras". Studia Mathematica, Tome 187 (2008) pp. 149-151. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm184-2-4/