An application of the Moore-Penrose inverse to antisymmetric relations

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.