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 $p_1,p_2<p$ 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 $\ell_2(\mathbb N)$.
Thanks in advance!
Best Answer
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 what 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 $$ E_{11}^{(2)}\geq E_{11}^{(4)}\geq E_{11}^{(8)}\geq\cdots $$