# is this von Neumann algebra a tensor product?

Let $H_1$ and $H_2$ be Hilbert spaces.
Let $A\subset B(H_1)$ be a factor 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.

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.