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Moore-Penrose inverse in rings with involution. (English) Zbl 1130.46032

Let \(R\) be a ring with involution. An element \(a\in R\) is called regular if there exists an element \(b\in R\) such that \(a=aba\). A regular element \(a \in R\) is called Moore-Penrose invertible if there is an element \(a^† \in R\) such that \(aa^† a=a\), \(a^† aa^†=a^†\), \((aa^†)^*=aa^†\) and \((a^† a)^*=a^† a\). An element \(a\in R\) is called left \(*\)-cancellable if \(a^*ax = a^*ay\) implies \(ax = ay\). An element \(a\in R\) is said to be well-supported if there exists a selfadjoint idempotent \(p\) such that \(ap = a\) and \(a^*a + 1 - p\) is regular (see Definition 6.5.3 of [B.Blackadar, ‘\(K\)-theory for operator algebras’ (Publications. Mathematical Sciences Research Institute 5, Cambridge University Press) (1998; Zbl 0913.46054)]).
The authors prove that an element \(a \in R\) is Moore–Penrose invertible if and only if \(a\) is left \(*\)-cancellable and well-supported. They present some applications of their results to the characterization of stable rank \(1\) and real rank \(1\) \(C^*\)-algebras. They algebraically prove reverse order rule \((ab)^†=b^† a^†\) under certain conditions and thus extend the known results for matrices [T.L.Boullion and P.L.Odell, ‘Generalized inverse matrices’ (New York etc.:Wiley–Interscience) (1971; Zbl 0223.15002)] and Hilbert space operators [R.H.Bouldin, Res.Notes Math.66, 233–249 (1982; Zbl 0522.47001)]. A reverse order rule for the weighted Moore–Penrose inverse in \(C^*\)-algebras is also obtained and therefore the matrix results of W.-N.Sun and Y.-M.Wei [SIAM J. Matrix Anal.Appl.19, No. 3, 772–775 (1998; Zbl 0911.15004)] are generalized.

MSC:

46L05 General theory of \(C^*\)-algebras
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
15A09 Theory of matrix inversion and generalized inverses
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References:

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