Fuglede theorem for symmetric spaces of $τ$-measurable operators

We extend the classical Fuglede commutativity theorem to the full scale of symmetrically normed operator ideals. Our main result provides a complete characterization: a symmetric ideal or symmetric operator space of $τ$-measurable operators satisfies the Fuglede theorem if and only if its commutative core has non-trivial Boyd indices, or equivalently, if it is an interpolation space in the scale of $L_p$-spaces for $1<p<\infty$. This criterion subsumes all previously known cases, including Lorentz and Schatten classes.

💡 Research Summary

The paper investigates the extension of the classical Fuglede–Putnam theorem from the setting of bounded normal operators on a Hilbert space to the much broader framework of symmetric spaces of τ‑measurable operators affiliated with a semifinite von Neumann algebra (M, τ). The classical theorem states that for normal operators A and B, the intertwining relation AT = TB implies A* T = TB*. The authors ask a quantitative version: if the commutator