A Remark on an Infinite Tensor Product of von Neumann Algebras

Abstract

Let $H_{\mathfrak{c}}$ be the incomplete infinite tensor product of Hilbert spaces $H_{⍳}$ containing a product vector $\otimes x_{⍳}$, where $\mathfrak{c}$ denotes the equivalence class of the ${\mathfrak{C}}_0$-sequence $\{x_{⍳}\}$. Let $E_{\mathfrak{c}}$ be the projection on $H_{\mathfrak{c}}$ in the complete infinite tensor product $H$ of $H_{⍳}$. Let $\mathfrak{R}$ be the von Neumann algebra on $H$ generated by von Neumann algebra ${\mathfrak{R}}_{⍳}$ on $H_{⍳}$ and $E(\mathfrak{c})$ be the central support of $E_{\mathfrak{c}}$ in ${\mathfrak{R}}^{\prime }$. Two ${\mathfrak{C}}_0$-sequences $\{x_{⍳}\}$ and $\{y_{⍳}\}$, and their equivalence classes $\mathfrak{c}$ and ${\mathfrak{c}}^{\prime }$, are defined to be $p$-equivalent if there exist partial isometries $p_{⍳}\in {\mathfrak{R}}_{⍳}^{\prime }$ such that $\{x_{⍳}\}$ and $\{p_{⍳}{y}_{⍳}\}$ are equivalent and $p_{⍳}^{\ast }{p}_{⍳}{y}_{⍳}=y_{⍳}$. They are defined to be _$u$-equivalent if $p_{⍳}$ can be chosen unitary. We prove that $E(\mathfrak{c})$ is the sum of $E_{{\mathfrak{c}}^{\prime }}$ with ${\mathfrak{c}}^{\prime }$, $p$-equivalent to $\mathfrak{c}$. If the index set is countable, $p$-equivalence and $u$-equivalence coincide.