On Type Classification of Factors Constructed as Infinite Tensor Products

(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