On relationship among three types of Birkhoff-James orthogonality
Soumitra Daptari, Koki Igarashi, Jumpei Nakamura, Ryotaro Tanaka
TL;DR
The paper investigates three Birkhoff-James orthogonality notions in Hilbert $\mathfrak{A}$-modules—strong $\perp_{BJ}^s$, quasi-strong $\perp_{BJ}^q$, and original $\perp_{BJ}$—and clarifies their relationships. It proves that $\perp_{BJ}^s$ and $\perp_{BJ}^q$ are equivalent on a full module precisely when the underlying C*-algebra $\mathfrak{A}$ is commutative, and that the equivalence of $\perp_{BJ}^q$ and $\perp_{BJ}$ implies $\mathfrak{A}$ is prime; the converse implications fail in general, as shown by examples involving primitive and non-simple algebras. The work also discusses the complexity of these equivalences, including a pair of concrete examples, and situates the results within broader classification themes for C*-algebras via Birkhoff-James orthogonality. Overall, it advances understanding of how algebraic structure governs orthogonality relations in Hilbert C*-modules and informs potential nonlinear classification approaches. $\perp_{BJ}^s$, $\perp_{BJ}^q$, $\perp_{BJ}$, commutativity, primeness, pure states, Hilbert $\mathfrak{A}$-modules$
Abstract
In this paper, we study three types of Birkhoff-James orthogonality in Hilbert $C^*$-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 $C^*$-module if and only if the underlying $C^*$-algebra is commutative, and that the equivalence of the quasi-strong and original Birkhoff-James orthogonality in a full Hilbert $C^*$-module implies the primeness of the underlying $C^*$-algebra. Moreover, two examples, explaining the complexity of conditions for full Hilbert $C^*$-modules in which the quasi-strong and original Birkhoff-James orthogonality are equivalent, are given in the $C^*$-algebra settings.