## An application of the Moore-Penrose inverse to antisymmetric relations

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- by Robert E. Hartwig
- Proc. Amer. Math. Soc.
**78**(1980), 181-186 - DOI: https://doi.org/10.1090/S0002-9939-1980-0550489-X
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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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## Bibliographic Information

- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**78**(1980), 181-186 - MSC: Primary 16A28; Secondary 15A09
- DOI: https://doi.org/10.1090/S0002-9939-1980-0550489-X
- MathSciNet review: 550489