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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An application of the Moore-Penrose inverse to antisymmetric relations

Author: Robert E. Hartwig
Journal: Proc. Amer. Math. Soc. 78 (1980), 181-186
MSC: Primary 16A28; Secondary 15A09
MathSciNet review: 550489
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Abstract: Let R be a star-ring and let $ {R_\dagger }$ denote the set of star-regular elements in R. It is shown that the relation $ a\Delta b$, defined by $ a{a^\ast}a = a{b^\ast}a$, is antisymmetric on $ {R_\dagger }$ provided that the two-term star-cancellation law and the positive-semidefinite axiom hold in R. This includes the star-regular elements of all $ {C^\ast}$-algebras, and in particular those elements in $ {{\mathbf{C}}_{n \times n}}$ and $ B(H)$, the bounded linear transformations on Hilbert space H.

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Article copyright: © Copyright 1980 American Mathematical Society