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!