On the converse of a theorem of Harte and Mbekhta: Erratum to ``On generalized inverses in C^*-algebras''

Abstract

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 , andThey 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 involution. 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 mistake 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.

On generalized inverses and Green’s relations

“…Combining theorem 2.5 (or corollary 2.9) and theorem 3.1, we then get directly the following existence criteria and commuting relations for the classical inverses [4], [2], [5], [6], [7], [8], [14], [9]:…”

“…There exist many specific generalized inverses in the literature, such as the group inverse, the Moore-Penrose inverse [1] or the Drazin inverse ( [2], [1]). Necessary and sufficient conditions for the existence of such inverses are known ([4], [2], [5], [6], [7], [8], [14], [9]), as are their properties. If one looks carefully at these results, it appears that these existence criteria all involve Green'relations [4], and that all inverses have double commuting properties.…”