Let be a magma and . The commutant of is defined as:
for all . We have that and for and that iff is abelian. We can also refer to the commutant of the whole structure as the center of (i.e. ).
Definition 1. A von Neumann algebra is a C*-algebra such that .
A projection in a *-algebra is an element such that . A partial isometry is an element such that is a projection. Recall an element is positive, denoted , if and , and that if . We will also say that two elements are orthogonal, denoted , if .
Definition 2. Two projections are Murray-von Neumann equivalent, denoted , if there is a partial isometry such that and . We say is subordinate to , denoted , if there is a projection such that and .
Proposition 3. Let be a *-algebra and be a sequence of pairs of projections such that and for and for all . Then Also, if for all then
Proof. If , then and Hence and . Now define and Then we have
where the last equality follows from orthogonality. Hence
Now suppose Then there are such that and Then since by the previous claim, it remains to show that But since for all So we have the result.
Proposition 4 (Schroder-Bernstein). Let and be projections in a *-algebra such that and Then .
It turns out that if a unital von Neumann algebra is a factor (), then is a total order on the projections.
Definition 5. A projection is
- abelian if is commutative;
- finite if where implies that ;
- infinite if it is not finite;
- properly infinite if and where and
Lemma 6. If is an infinite projection in a von Neumann algebra , then there is a projection such that is nonzero and properly infinite.
This allows for a somewhat complicated decomposition (see  for details) of
where is a discrete central projection, is the largest finite continuous central projection, is the largest properly infinite semifinite continuous projection, and is a purely infinite projection. The algebra is said to be of pure type if for all
 Blackadar, Bruce. Operator Algebras. Encyclopedia of Mathematical Sciences. Springer-Verlag. 2006.