A Remark on an Infinite Tensor Product of von Neumann Algebras

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.