STUDIA MATHEMATICA 184 (2) (2008) On the converse of a theorem of Harte and Mbekhta: Erratum to “On generalized inverses in C * -algebras” by Xavier Mary (Malakoff) Abstract. 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 ∈ A has a commuting generalized inverse then it is decomposably regular , and A = aA + a -1 (0) with aA ∩ a -1 (0) = {0}, (1) A = Aa + a -1 (0) with Aa ∩ a -1 (0) = {0}. (2) They then write “The conditions (1) and (2) are not together sufficient for a ∈ aAa to be simply polar” (i.e. to have a commuting generalized inverse) and they exhibit a counterexample. The latter sentence is false, for their conditions actually imply simple polarity of a: Theorem 2. Let A be a monoid (semigroup with identity ) with involu- tion. Then the following conditions are equivalent : 1. a ∈ A is simply polar. 2. Aa = Aa 2 and aA = a 2 A. Note that the latter conditions are weaker than those in [3] (just multiply (1) on the left by a and (2) on the right by a), and that A need not be a ring. Before giving the proof of the theorem, let us describe the original mis- take of Harte and Mbekhta. It is not true that both conditions (1) and (2) ((9.1) and (9.2) in [3]) are satisfied by the example on page 75, lines 6 to 4 from the bottom, because if they were, then the example would satisfy the relations Aa = Aa 2 and 2000 Mathematics Subject Classification : Primary 46L05, 20M99. Key words and phrases : generalized inverse, group inverse. [149] c  Instytut Matematyczny PAN, 2008