<p>In this paper, we study three types of Birkhoff–James orthogonality in Hilbert <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-module if and only if the underlying <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebra is commutative, and that the equivalence of the quasi-strong and original Birkhoff–James orthogonality in a full Hilbert <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-module implies the primeness of the underlying <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebra. Moreover, two examples, explaining the complexity of conditions for full Hilbert <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-modules in which the quasi-strong and original Birkhoff–James orthogonality are equivalent, are given in the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$C^*$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebra settings.</p>

On the relationship among three types of Birkhoff–James orthogonality

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2026年7月30日

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Soumitra Daptari,
Koki Igarashi,
Jumpei Nakamura,
Ryotaro Tanaka

摘要

In this paper, we study three types of Birkhoff–James orthogonality in Hilbert $C^*$ $C^{*}$ -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 $C^*$ $C^{*}$ -module if and only if the underlying $C^*$ $C^{*}$ -algebra is commutative, and that the equivalence of the quasi-strong and original Birkhoff–James orthogonality in a full Hilbert $C^*$ $C^{*}$ -module implies the primeness of the underlying $C^*$ $C^{*}$ -algebra. Moreover, two examples, explaining the complexity of conditions for full Hilbert $C^*$ $C^{*}$ -modules in which the quasi-strong and original Birkhoff–James orthogonality are equivalent, are given in the $C^*$ $C^{*}$ -algebra settings.