Relative positions of half-sided modular inclusions
Ian Koot · Letters in Mathematical Physics · 2025
Abstract Let $$K_1 \subset H$$ K 1 ⊂ H and $$K_2 \subset H$$ K 2 ⊂ H be half-sided modular inclusions in a common standard subspace H . We prove that the inclusion $$K_1 \subset K_2$$ K 1 ⊂ K 2 holds if and only if we have an inclusion of spectral subspaces of the generators of the positive one-parameter groups associated to the half-sided modular inclusions $$K_1 \subset H$$ K 1 ⊂ H and $$K_2 \subset H$$ K 2 ⊂ H . From this we give a characterization of this situation in terms of (operator-valued) symmetric inner functions. We illustrate these characterizations with some examples of non-trivial phenomena occurring in this setting.