On generalized inverses in C*-algebras

On the Moore-Penrose inverse in rings with involution

“…In a unital C -algebra A, since a 2 A is Moore-Penrose invertible if and only if af1g ¤ ¿ (see [3]), we get the following result as a consequence of Theorem 4.…”

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

“…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. …”

“…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.…”

“…In [3], Harte and Mbekhta give the following theorem (A is a C * -algebra): Theorem 1. A normalized commuting inverse is unique.…”

Equalities of ideals associated with two projections in rings with involution

“…Let us recall that a ring R with involution has the Gelfand-Naimark property if 1 + x * x ∈ R −1 for all x ∈ R. It is known that any C * -algebra has the Gelfand-Naimark property. See also [6] and [11].…”