lattice of projections
The set of projections in , although not forming a vector space, has a very rich structure. In this entry we are going to endow this set with a partial ordering in a that it becomes a complete lattice. The lattice structure of the set of projections has profound consequences on the structure of von Neumann algebras.
1 The Lattice of Projections
In Hilbert spaces there is a bijective correspondence between closed subspaces and projections (see this entry (http://planetmath.org/ProjectionsAndClosedSubspaces)). This correspondence is given by
where is a projection and denotes the range of .
But since projections are self-adjoint operators (in fact they are positive operators, as ), they inherit the natural partial ordering of self-adjoint operators (http://planetmath.org/OrderingOfSelfAdjoints), which we denote by , and whose definition we recall now
As the following theorem shows, these two orderings coincide. Thus, we shall not make any more distinctions of notation between them.
Theorem 1 - Let be projections in . The following conditions are equivalent:
Two closed subspaces in have a greatest lower bound and a least upper bound . Specifically, is precisely the intersection and is precisely the closure of the subspace generated by and . Hence, if are projections in then is the projection onto and is the projection onto the closure of .
The above discussion clarifies that the set of projections in has a lattice structure. In fact, the set of projections forms a complete lattice, by somewhat as above:
Every family of closed subspaces in possesses an infimum and a supremum , which are, respectively, the intersection of all and the closure of the subspace generated by all . There is, of course, a correspondent in terms of projections: every family of projections has an infimum and a supremum , which are, respectively, the projection onto the intersection of all and the projection onto the closure of the subspace generated by all .
2 Additional Lattice Features
Projections on one-dimensional subspaces are usually called minimal projections and they are in fact minimal in the sense that: there are no closed subspaces strictly between and a one-dimensional subspace, and every closed subspace other than contains a one-dimensional subspace. This means that the lattice of projections in is an atomic lattice and its atoms are precisely the projections on one-dimensional subspaces.
Moreover, every closed subspace of is the closure of the span of its one-dimensional subspaces. Thus, the lattice of projections in is an atomistic lattice.
We shall see further ahead in this entry, when we discuss orthogonal projections, that the lattice of projections in is an orthomodular lattice.
3 Commuting and Orthogonal Projections
When two projections commute, the projections and can be described algebraically in a very . We shall see at the end of this section that and commute precisely when its corresponding subspaces and are ”perpendicular”.
Theorem 2 - Let be commuting projections (i.e. ), then
. In particular, is closed.
Two projections are said to be orthogonal if . This is equivalent to say that its corresponding subspaces are orthogonal ( lies in the orthogonal complement of ).
Corollary 1 - Two projections are orthogonal if and only if . When this is so, then .
Corollary 2 - Let be projections in such that . Then is the projection onto .
We can now see that commute if and only if and are ”perpendicular”. A somewhat informal and intuitive definition of ”perpendicular” is that of requiring the two subspaces to be orthogonal outside their intersection (this is different of , since orthogonal subspaces do not intersect each other). More rigorously, and commute if and only if the subspaces and are orthogonal.
This can be proved using all the above results: The two subspaces are orthogonal iff
We can now also see that the lattice of projections is orthomodular: Suppose . Then, using the above results,
4 Nets of Projections
In the following we discuss some useful and interesting results about convergence and limits of projections.
Let be a poset. A net of projections is said to be increasing if . Decreasing nets are defined similarly.
Theorem 3 - Let be an increasing net of projections. Then for every .
Similarly for decreasing nets of projections,
Theorem 4 - Let be a decreasing net of projections. Then for every .
In other words, converges to in the strong operator topology.
Theorem 5 - Let be a set and be a family of pairwise orthogonal projections. Then is summable and for all .
|Title||lattice of projections|
|Date of creation||2013-03-22 17:53:29|
|Last modified on||2013-03-22 17:53:29|
|Last modified by||asteroid (17536)|
|Synonym||projections in Hilbert spaces|