A Remark on an Infinite Tensor Product of von Neumann Algebras
Huzihiro Araki
Kyoto University, JapanYoshiomi Nakagami
Tokyo Institute of Technology, Japan

Abstract
Let Hc be the incomplete infinite tensor product of Hilbert spaces H⍳ containing a product vector ⊗_x_, where c denotes the equivalence class of the ℭ0-sequence {x⍳} . Let Ec be the projection on Hc in the complete infinite tensor product H of H⍳. Let ℜ be the von Neumann algebra on H generated by von Neumann algebra ℜ⍳ on H⍳ and E(c) be the central support of Ec in ℜ'. Two ℭ0-sequences {x⍳} and {y⍳}, and their equivalence classes c and c', are defined to be p-equivalent if there exist partial isometries _p⍳_∈ ℜ'⍳ such that { x⍳ } and {pcy⍳} are equivalent and p*⍳pcy⍳= yc. They are defined to be u-equivalent if p⍳ can be chosen unitary. We prove that E(c) is the sum of Ec' with c', p-equivalent to c. If the index set is countable, p-equivalence and u-equivalence coincide.
Cite this article
Huzihiro Araki, Yoshiomi Nakagami, A Remark on an Infinite Tensor Product of von Neumann Algebras. Publ. Res. Inst. Math. Sci. 8 (1972), no. 2, pp. 363–374
DOI 10.2977/PRIMS/1195193114