Let $H\_1$ and $H\_2$ be Hilbert spaces.  
 Let $A\subset B(H\_1)$ be a [factor](http://en.wikipedia.org/wiki/Von_Neumann_algebra#Factors) and $A'$ its commutant.


If a von Neumann algebra $M\subset B(H\_1\otimes H\_2)$ contains $A\otimes 1$ and commutes with $A'\otimes 1$, is it then necessarily of the form $M=A\otimes B$ for some von Neumann algebra $B\subset B(H\_2)$?


I think that I know how to prove this if $A$ is hyperfinite, and I wonder if it's true in general.



is this von Neumann algebra a tensor product?
What about $A = A\_1 \oplus A\_2$ and $M = (A\_1\otimes 1) \oplus (A\_2 \otimes B(H\_2))$?


It seems like your condition just says that $A\otimes 1 \subseteq M \subseteq A \otimes B(H\_2)$, did you leave something out?


Edit: in case $A$ is a factor, the answer is yes. Ge and Kadison, On tensor products of von Neumann algebras, *Invent. Math.* **123** (1996), 453-466.


