Editing Von Neumann algebra (section)

== Projections ==
Operators ''E'' in a von Neumann algebra for which ''E'' = ''EE'' = ''E*'' are called '''projections'''; they are exactly the operators which give an orthogonal projection of ''H'' onto some closed subspace. A subspace of the Hilbert space ''H'' is said to '''belong to''' the von Neumann algebra ''M'' if it is the image of some projection in ''M''. This establishes a 1:1 correspondence between projections of ''M'' and subspaces that belong to ''M''. Informally these are the closed subspaces that can be described using elements of ''M'', or that ''M'' "knows" about.

It can be shown that the closure of the image of any operator in ''M'' and the kernel of any operator in ''M'' belongs to ''M''. Also, the closure of the image under an operator of ''M'' of any subspace belonging to ''M'' also belongs to ''M''.  (These results are a consequence of the [[polar decomposition]]).

===Comparison theory of projections===
The basic theory of projections was worked out by {{harvtxt|Murray|von Neumann|1936}}. Two subspaces belonging to ''M'' are called ('''Murray–von Neumann''') '''equivalent''' if there is a partial isometry mapping the first isomorphically onto the other that is an element of the von Neumann algebra (informally, if ''M'' "knows" that the subspaces are isomorphic). This induces a natural [[equivalence relation]] on projections by defining ''E'' to be equivalent to ''F'' if the corresponding subspaces are equivalent, or in other words if there is a [[partial isometry]] of ''H'' that maps the image of ''E'' isometrically to the image of ''F'' and is an element of the von Neumann algebra. Another way of stating this is that ''E'' is equivalent to ''F'' if ''E=uu*'' and ''F=u*u'' for some partial isometry ''u'' in ''M''.

The equivalence relation ~ thus defined is additive in the following sense: Suppose ''E''<sub>1</sub> ~ ''F''<sub>1</sub> and ''E''<sub>2</sub> ~ ''F''<sub>2</sub>. If ''E''<sub>1</sub> ⊥ ''E''<sub>2</sub> and ''F''<sub>1</sub> ⊥ ''F''<sub>2</sub>, then  ''E''<sub>1</sub> + ''E''<sub>2</sub> ~ ''F''<sub>1</sub> + ''F''<sub>2</sub>. Additivity would ''not'' generally hold if one were to require unitary equivalence in the definition of ~, i.e. if we say ''E'' is equivalent to ''F'' if ''u*Eu'' = ''F'' for some unitary ''u''. The [[Schröder–Bernstein theorems for operator algebras]] gives a sufficient condition for Murray-von Neumann equivalence. 

The subspaces belonging to ''M'' are partially ordered by inclusion, and this induces a partial order ≤ of projections. There is also a natural partial order on the set of ''equivalence classes'' of projections, induced by the partial order ≤ of projections. If ''M'' is a factor, ≤ is a total order on equivalence classes of projections, described in the section on traces below.

A projection (or subspace belonging to ''M'') ''E'' is said to be a '''finite projection''' if there is no projection ''F'' &lt; ''E'' (meaning ''F'' &le; ''E''  and ''F'' &ne; ''E'') that is equivalent to ''E''.  For example, all finite-dimensional projections (or subspaces) are finite (since isometries between Hilbert spaces leave the dimension fixed), but the identity operator on an infinite-dimensional Hilbert space is not finite in the von Neumann algebra of all bounded operators on it, since it is isometrically isomorphic to a proper subset of itself. However it is possible for infinite dimensional subspaces to be finite.

Orthogonal projections are noncommutative analogues of indicator functions in ''L''<sup>∞</sup>('''R'''). ''L''<sup>∞</sup>('''R''') is the ||·||<sub>∞</sub>-closure of the subspace generated by the indicator functions. Similarly, a von Neumann algebra is generated by its projections; this is a consequence of the [[Self-adjoint operator#Spectral theorem|spectral theorem for self-adjoint operators]].

The projections of a finite factor form a [[continuous geometry]].