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(Hv, ev) is formed. Suppose further that a von Neumann algebra Mv be given on each Hv. Each operator T in Mv is extended over all of H, which will be denoted as T. All these operators T together form a von Neumann algebra on H, which we write as Mv. The von Neumann algebra M on H generated by these Mv's (z/ = l, 2, • • • ) is what we call as the infinite tensor product of Mv. Suppose now that each Mv given is a factor of type I. Then, Hv can be decomposed as a direct product of two Hilbert spaces HV1, HV 2 : HV = HV1®HV2, and Mv is thereby identified with B(R^)®IV2, where B(HV1) is the total operator algebra on HV1 and JV2 is a von Neumann algebra consisting of all scalar multiples of the identity operator on HV2. With respect to this direct product decomposition, 0V can be expressed as

On Type Classification of Factors Constructed as Infinite Tensor Products