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.

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