If $\mathcal{H}$ is a separable Hilbert space then we may decompose a von Neumann algebra $\mathcal{M}\subset B(\mathcal{H})$ into a direct integral of factors, therefore one considers factors as the building block of von Neumann algebras and focuses on factors when classifying them.
My question is: Why are factors the natural building block of a von Neumann algebra (despite that it finally works)?
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What Murray-von Neumann did was to show that there is an infinite-dimensional generalization of the following fact.
If $\mathcal H$ is finite-dimensional and $\mathcal M\subset\mathcal B(\mathcal H)$ is a von Neumann algebra, it is a basic exercise that we can see $\mathcal B(\mathcal H)$ as $M_n(\mathbb C)$ for $n=\dim\mathcal H$. And in that situation, $\mathcal M$ is isomorphic to $$\tag{*}\bigoplus_{k=1}^{\ell} M_{n(k)}(\mathbb C),$$ where the blocks are given by the minimal central projections (i.e., each block is $P\mathcal M P$, with $P$ a minimal central projection.
When $\mathcal H$ is infinite-dimensional, the same idea works. Thing is, now the centre may not have minimal projections, but what they proved is that there exists a Borel space $X$ and a Borel measure $\mu$ such that $$ \mathcal M=\int_X^{\oplus} \mathcal M_\lambda\,d\mu(\lambda), $$ where the function $\lambda\longmapsto \mathcal M_\lambda$ is factor-valued a.e.
In the particular case when $X$ is finite, $\mu$ is the counting measure and the $\mathcal M_\lambda$ are finite-dimensional, one recovers $(*)$.
To understand any category of algebraic objects (groups, rings, etc.) one of the key goals is to understand the simple objects (the ones, $M$, for which all morphisms $M\rightarrow N$ for any object $N$ is injective). For von Neuman algebras and normal homomorphisms the simple objects are the factors.