# Visualizing projections in type $II_1$ AFD von Neumann algebras

I'm having a lot of trouble to picture the elements of the AFD (hyperfinite) $$II_1$$ von Neumann algebra. I would like to see concrete examples of operators and projections belonging to the hyperfinite $$II_1$$ factor $$R$$ when the when it's viewed as a subalgebra of $$B(H)$$ (assuming that this inclusion is possible).

For now, I'd like to make concrete the fact that $$II_1$$ algebras are diffuse, i.e. have no minimal projections. I'm trying to see how a projection $$p>0$$ could be decomposed in two other projections $$p_1,p_2 with $$p=p_1+p_2$$ and also how these projections could be approximated by the finite subalgebras.

When I try to follow the $$II_1$$ factor constructions I get lost in the GNS procedure. Also, when trying to use the $$M_{2^n}$$ construction, I'm not sure how the finite subalgebras belong to the hyperfinite factor. The naive visualization of finite algebras of type $$I_{n}$$ in $$L(H)$$ takes me to finite matrix algebras which do have minimal projections. I don't know where I'm making the mistakes.

I am overwhelmed by the loads of new concepts in the von neumann algebra theory.

I would highly appreciate any hints or references on how the operators and projections in the hyperfinite factor could be made explicity in some $$B(H)$$, perhaps operators in $$\ell_2(\mathbb N)$$.

The inclusion into $$B(H)$$ for some $$H$$ is always possible. Generally one defines a vN-algebra as a self-adjoint subalgebra of $$B(H)$$ which is WOT-closed. Alternatively, vN-algebras are C*-algebras, and these can be embedded into $$B(H)$$ via the GNS construction.
The fact that a $$II_1$$ factor is diffuse follows from the fact that a factor $$M$$ has a minimal projection if and only if $$M \simeq B(H)$$ for some Hilbert space $$H$$. I'll leave this as a fact, but feel free to ask about clarification about the proof.
Clearly any $$II_1$$ factor has a (faithful) tracial state and is infinite dimensional, so it cannot be $$B(H)$$ ($$B(H)$$ has no tracial state when $$H$$ is infinite-dimensional; in fact, not even the compacts do). So it has to be diffuse. Consider $$A = M_{2^{\infty}}$$ (the direct limit $$\underset{\to}{\lim} M_{2^n}$$ with connecting maps $$a \mapsto a \otimes 1$$). This is a uniformly hyperfinite C*-algebra, and has a unique faithful trace $$\tau$$. The hyperfinite $$II_1$$ factor comes from taking the GNS representation with respect to $$\tau$$: $$M = \pi_\tau(A)'' = \overline{\pi_\tau(A)}^{\text{SOT}} \subseteq B(L^2(A,\tau))$$, where $$(a\xi,b\xi) = \tau(b^*a)$$ for $$a,b \in A$$, and $$\xi$$ is a unit vector which is separating and cyclic. The trace on $$M$$ is given by $$a \mapsto (a\xi,\xi)$$.
I'll give an explicit example of a projection in $$A \subseteq M$$ which we can decompose in the way you want. $$A$$ can be thought of as the infinite tensor product of $$M_2$$. So $$A = \otimes_1^{\infty}M_2$$, which is really the norm closure of $$\cup_n (M_2^{\otimes n}\otimes 1\otimes\cdots)$$. Let $$p = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \otimes 1 \otimes\cdots.$$ We can write this projection in the following way. Let $$p_1 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \otimes \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \otimes 1 \otimes\cdots$$ and $$p_2 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \otimes \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \otimes 1 \otimes\cdots.$$ Now its clear that $$p_1,p_2 < p$$ and that $$p = p_1 + p_2$$. Edit: Note that you can keep on doing this, so these projections can't be minimal!
This is a "less naive" way of seeing the chain $$M_2(\mathbb C)\subset M_4(\mathbb C)\subset\cdots\subset B(H)$$ for infinite-dimensional separable $$H$$ (and it is was was done in PStheman's answer, just a bit more explicit here).
You see $$M_2(\mathbb C)$$ as $$\begin{bmatrix} a&b\\ c&d\\ &&a&b\\ &&c&d\\ &&&&a&b\\ &&&&c&d\\ &&&&&&a&b\\ &&&&&&c&d\\ &&&&&&&&\ddots \end{bmatrix},$$ then $$M_4(\mathbb C)$$ as $$\begin{bmatrix} a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ a_{41}&a_{42}&a_{43}&a_{44}\\ &&&&a_{11}&a_{12}&a_{13}&a_{14}\\ &&&&a_{21}&a_{22}&a_{23}&a_{24}\\ &&&&a_{31}&a_{32}&a_{33}&a_{34}\\ &&&&a_{41}&a_{42}&a_{43}&a_{44}\\ &&&&&&&&\ddots \end{bmatrix}.$$ So for example take $$E_{11}^{(2)}\in M_2(\mathbb C)$$, and let's find subprojections of it: $$E_{11}^{(2)}=\begin{bmatrix} 1\\ &0\\ &&1\\ &&&0\\ &&&&1\\ &&&&&0\\ &&&&&&1\\ &&&&&&&0\\ &&&&&&&&\ddots \end{bmatrix}.$$ Now you can see that $$E_{11}{(4)}$$ is a subprojection of $$E_{11}^{(2)}$$: $$E_{11}^{(4)}=\begin{bmatrix} 1\\ &0\\ &&0\\ &&&0\\ &&&&1\\ &&&&&0\\ &&&&&&0\\ &&&&&&&0\\ &&&&&&&&\ddots \end{bmatrix}.$$ Continuing this way you can get the proper chain of projections $$E_{11}^{(2)}\geq E_{11}^{(4)}\geq E_{11}^{(8)}\geq\cdots$$