Let $B$ be a von-Neumann algebra and let $M$ be a right-Hilbert-$B$-module and let $N$ be a left-Hilbert-$B$-module. In this situation, Connes' fusion product $M \boxtimes_B N$ of $M$ and $N$ over $B$ is defined, which has a somewhat complicated definition using the structure of von-Neumann algebras.
My question is: What is wrong with the following definition of a tensor product of $M$ and $N$ over $B$?
Set $$ M \otimes_B N := M \hat{\otimes}_{\mathbb{C}} N ~/~ \overline{\mathcal{U}}, $$ where $\overline{\mathcal{U}}$ is the closure of the space $$\mathcal{U} := \mathrm{span}\{ m\cdot b \otimes n - m \otimes b \cdot n \mid m \in M, n \in N, b \in B\}$$ inside the Hilbert space tensor product of $M$ and $N$ over $\mathbb{C}$. This is the same thing as $$M \otimes_B N := \mathrm{HS}_A(M^*, N) $$ the subspace of Hilbert-Schmidt operators from $M^*$ to $N$ that commute with the action of $A$.
