Visualizing projections in type $II_1$ AFD von Neumann algebras

Question

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!

Answer 1

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 $$ E_{11}^{(2)}\geq E_{11}^{(4)}\geq E_{11}^{(8)}\geq\cdots $$

Answer 2

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!