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, a \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$ .