# von Neumann Algebras

Let $M$ be a magma and $S\subseteq M$.  The commutant of $S$ is defined as:

$S'=\{x\in M: sx=xs\}$

for all $s\in S$.  We have that $S'=S^{(2n+1)}$ and $S''=S^{(2n)}$ for $1\leq n\in\mathbb{N}$ and that $M=M'$ iff $M$ is abelian.  We can also refer to the commutant of the whole structure $M$ as the center of $M$ (i.e. $M'=Z(M)$).

Definition 1.  A von Neumann algebra is a C*-algebra $A$ such that $A=A''$.

A projection in a *-algebra is an element $p$ such that $p^2=p=p^*$.  A partial isometry is an element $u$ such that $u^*u$ is a projection. Recall an element $x$ is positive, denoted $x\geq 0$, if $x^*=x$ and $\sigma_A(x)\subseteq [0,\infty)$, and that $x\leq y$ if $y-x\geq 0$.  We will also say that two elements $x,y$ are orthogonal, denoted $x\perp y$, if $xy=0$.

Definition 2.  Two projections $p,q$ are Murray-von Neumann equivalent, denoted $p\sim q$, if there is a partial isometry $u$ such that $p=u^*u$ and $q=uu^*$.  We say $p$ is subordinate to $q$, denoted $p\preceq q$, if there is a projection $q'$ such that $p\sim q'$ and $q'\leq q$.

Proposition 3.  Let $A$ be a *-algebra and $\{(p_i,q_i)\}$ be a sequence of pairs of projections such that $p_i\perp p_j$ and $q_i\perp q_j$ for $i\neq j$ and $p_i\sim q_i$ for all $i$.  Then $\sum p_i\sim\sum q_i.$  Also, if $p_i\preceq q_i$ for all $i,$ then $\sum p_i\preceq\sum q_i.$

Proof.  If $p_i\sim q_i$, then $p_i=u_i^*u_i$ and $q_i=u_iu_i^*.$  Hence $\sum p_i=\sum u_i^*u_i$ and $\sum q_i=u_i^*u_i$.  Now define $u=\sum u_i$ and $u^*=\sum u_i^*.$  Then we have

$\displaystyle u^*u=\left(\sum u_i^*\right)\left(\sum u_i\right)=\sum u_i^*u_j=\sum u_i^*u_i$

where the last equality follows from orthogonality.  Hence $\sum p_i\sim\sum q_i.$

Now suppose $p_i\preceq q_i.$  Then there are $q_i'$ such that $p_i\sim q_i'$ and $q_i'\leq q_i.$  Then since $\sum p_i\sim\sum q_i'$ by the previous claim, it remains to show that $\sum q_i'\leq\sum q_i.$  But $\sum q_i-\sum q_i'=\sum (q_i-q_i')\geq 0$ since $q_i'\leq q_i$ for all $i.$  So we have the result.

Proposition 4 (Schroder-Bernstein).  Let $p$ and $q$ be projections in a *-algebra such that $p\preceq q$ and $q\preceq p.$  Then $p\sim q$.

It turns out that if a unital von Neumann algebra $A$ is a factor ($Z(M)=1$), then $\preceq$ is a total order on the projections.

Definition 5.  A projection $p\in A$ is

1. abelian if $pAp$ is commutative;
2. finite if $p\sim p'$ where $p'\leq p$ implies that $p=p'$;
3. infinite if it is not finite;
4. properly infinite if $p\sim p_1$ and $p\sim p_2$ where $p_1,p_2\leq p$ and $p_1\perp p_2.$

Lemma 6.  If $p$ is an infinite projection in a von Neumann algebra $A$, then there is a projection $z\in Z(A)$ such that $pz$ is nonzero and properly infinite.

This allows for a somewhat complicated decomposition (see [1] for details) of $A$

$A=Az_1\oplus Az_{2_1}\oplus Az_{2_\infty}\oplus Az_3$

where $z_1$ is a discrete central projection, $z_{2_1}$ is the largest finite continuous central projection, $z_{2_\infty}$ is the largest properly infinite semifinite continuous projection, and $z_3$ is a purely infinite projection.  The algebra $A$ is said to be of pure type $\alpha$ if $z_\beta=0$ for all $\beta\neq\alpha.$

[1]  Blackadar, Bruce.  Operator Algebras.  Encyclopedia of Mathematical Sciences.  Springer-Verlag.  2006.