On the relationship among three types of Birkhoff–James orthogonality

Abstract In this paper, we study three types of Birkhoff–James orthogonality in Hilbert $$C^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -modules, that is, the strong, quasi-strong, and original Birkhoff–James orthogonality. In general, the strong Birkhoff–James orthogonality is stronger than the quasi-strong Birkhoff–James orthogonality, and the quasi-strong Birkhoff–James orthogonality is stronger than the original Birkhoff–James orthogonality. Meanwhile, each reverse implication in this chain requires additional conditions. As the main results, we show that the strong and quasi-strong Birkhoff–James orthogonality are equivalent in a full Hilbert $$C^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -module if and only if the underlying $$C^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -algebra is commutative, and that the equivalence of the quasi-strong and original Birkhoff–James orthogonality in a full Hilbert $$C^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -module implies the primeness of the underlying $$C^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -algebra. Moreover, two examples, explaining the complexity of conditions for full Hilbert $$C^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -modules in which the quasi-strong and original Birkhoff–James orthogonality are equivalent, are given in the $$C^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -algebra settings.