# A Remark on an Infinite Tensor Product of von Neumann Algebras

### Huzihiro Araki

Kyoto University, Japan### Yoshiomi 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.