I was reading the section on the structure of type I von Neumann algebras in John B. Conway's "A course in operator theory" and a few questions about certain definitions and references arose, I was just seeking some clarification :).

Let $A$ be a $C^*$-algebra, $M_n(A)$ is the algebra of $n \times n$ matrices with entries in $A$, $A^{(n)}$ is the direct sum of $A$ with itself $n$ times.

On page 285 he defines:

"**50.10 Definition.** *If $\mathcal{A}$ is a von Neumann algebra and $n$ is some cardinal number, let
$M_n(\mathcal{A}) = \{ (\mathcal{A}')^{(n)} \}'$*."

**1 - What does this $M_n(\mathcal{A})$ have to do with the algebra of $n \times n$ matrices defined above?**

Then on the next page (page 286) he writes:

"*Another way to define $M_n(\mathcal{A})$ is to define the tensor product of von Neumann algebras. In this setting $M_n(\mathcal{A})$ is precisely $\mathcal{A}\otimes \mathcal{B}(\mathcal{K})$, where $\mathcal{K}$ is a Hilbert space with dimension $n$. In fact the Hilbert space $\mathcal{H} \otimes \mathcal{K}$ can be identified with $\mathcal{H}^{(n)}$. With this identification , $\mathcal{A}' \otimes \mathbb{C} \subseteq \mathcal{B}(\mathcal{H} \otimes \mathcal{K})$ is identified with $\mathcal{A}'^{(n)}$ and $M_n(\mathcal{A})$ with $\mathcal{A} \otimes \mathcal{B}(\mathcal{K})$. The reader can see the references for an exposition of the tensor product of von Neumann algebras.*"

I couldn't find the reference for this so I wanted to ask:

**2 - In this paragraph, is this supposed to apply only to type I algebras?**

**3 - What does it mean for $\mathcal{A}' \otimes \mathbb{C} \subseteq \mathcal{B}(\mathcal{H} \otimes \mathcal{K})$ to be "identified" with $\mathcal{A}'^{(n)}$ and $M_n(\mathcal{A})$ with $\mathcal{A} \otimes \mathcal{B}(\mathcal{K})$? Does that mean that $\mathcal{A}' \otimes \mathbb{C}$ is isomorphic to $\mathcal{A}'^{(n)}$ and $M_n(\mathcal{A})$ is isomorphic to $\mathcal{A} \otimes \mathcal{B}(\mathcal{K})$?**

Does anyone know? Or know where I can find a reference for that last paragraph?