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 (£0-sequence {x ( }. Let EC be the projection on He in the complete infinite tensor product H of Ht. Let Sft be the von Neumann algebra on H generated by von Neumann algebra 3ft, on Hc and E(c) be the central support of Ef in 3ft'. Two (£0-sequences {x t} and {ye}, and their equivalence classes c and c', are defined to be p-equivalent if there exist partial isometries p,e3ft' such that {x,} and {peye} are equivalent and p*pcyc = yc- They are defined to be u-equivalent if pc can be chosen unitary. We prove that E(c) is the sum of Ec* with c', pequivalent to c. If the index set is countable, p-equivalence and &-equivalence coincide.