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The factorizable vectors of a complete Boolean algebra of type I factors, acting on a separable Hilbert space, are shown to be total, resolving a conjecture of Araki and Woods. En route, the spectral theory of noise-type Boolean algebras of Tsirelson is cast in the noncommutative language of "factorizations with unit" for which a multi-layered characterization of being "of Fock type" is provided.
Matija Vidmar (2024) studied this question.