A partial classification into unitary equivalence classes of complete Boolean algebras of type I factors is given. Any complete atomic Boolean algebra of type I factors is unitarily equivalent to a discrete tensor product of type I factors. We establish a one-to-one correspondence between the unitary equivalence classes of complete nonatomic Boolean algebras of type I factors satisfying a certain condition, and the unitary equivalence classes of complete nonatomic Boolean algebras of projections. A continuous tensor product of Hilbert spaces is defined which is a generalization of the discrete infinite incomplete tensor product space defined by von Neumann. On a separable Hilbert space, any complete nonatomic Boolean algebra of type I factors satisfying the previously mentioned condition is unitarily equivalent to a continuous tensor product of type I factors. An application to the representations of the canonical commutation relations of quantum field theory is made.
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Huzihiro Araki, E. J. Woods, Complete Boolean algebras of type I factors. Publ. Res. Inst. Math. Sci. 2 (1966), no. 2, pp. 157–242DOI 10.2977/PRIMS/1195195888