Bell inequalities and operator algebras

Quantum Bell-type inequalities are defined in terms of two (or more) subsys-tems of a quantum system. The subsystems may be treated either via (local) Hilbert spaces, — tensor factors of the given (global) Hilbert space, or via commuting (local) operator algebras. The latter approach is less restrictive, it just requires that the given operators commute whenever they belong to different subsystems. Are these two approaches equivalent? Background Quantum Bell-type inequalities are discussed in Problem 26 “Bell inequalities holding for all quantum states ” and Partial Solutions to Problem 1 “All the Bell inequalities ” (and many other texts, including my own [1]) without a clear indication of the approach used. Some relevant formal definitions (for two subsystems) may be found in [1], Sect. 2 (and Sect. 1). Formulas (2.1–2.3) of [1] formalize the less restrictive approach (via commuting operators), while formulas (2.4–2.6) formalize