lattice of projections (original) (raw)

Recall that a projection P in B⁢(H) is a bounded (http://planetmath.org/BoundedOperator) self-adjoint operator satisfying P2=P.

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 P is a projection and Ran⁢(P) denotes the range of P.

Since the set of closed subspaces can be partially ordered by inclusion, we can define a partial order ≤ in the set of projections using the above correspondence:

But since projections are self-adjoint operators (in fact they are positive operators, as P=P*⁢P), they inherit the natural partial ordering of self-adjoint operators (http://planetmath.org/OrderingOfSelfAdjoints), which we denote by ≤s⁢a, 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 P,Q be projections in B⁢(H). The following conditions are equivalent:

• •
  Ran⁢(P)⊆Ran⁢(Q) (i.e. P≤Q)
• •
  Q⁢P=P
• •
  P⁢Q=P
• •
  ∥P⁢x∥≤∥Q⁢x∥ for all x∈H
• •
  P≤s⁢aQ

Two closed subspaces Y,Z in H have a greatest lower bound Y∧Z and a least upper bound Y∨Z. Specifically, Y∧Z is precisely the intersection Y∩Z and Y∨Z is precisely the closure of the subspace generated by Y and Z. Hence, if P,Q are projections in B⁢(H) then P∧Q is the projection onto Ran⁢(P)∩Ran⁢(Q) and P∨Q is the projection onto the closure of Ran⁢(P)+Ran⁢(Q).

The above discussion clarifies that the set of projections in B⁢(H) has a lattice structure. In fact, the set of projections forms a complete lattice, by somewhat as above:

Every family {Yα} of closed subspaces in H possesses an infimum ⋀Yα and a supremum ⋁Yα, which are, respectively, the intersection of all Yα and the closure of the subspace generated by all Yα. There is, of course, a correspondent in terms of projections: every family {Pα} of projections has an infimum ⋀Pα and a supremum ⋁Pα, which are, respectively, the projection onto the intersection of all Ran⁢(Pα) and the projection onto the closure of the subspace generated by all Ran⁢(Pα).

2 Additional Lattice Features

• •
• •
  Also, it is modular if and only if H is finite dimensional. Nevertheless, there are important of von Neumann algebras (a particular type of subalgebras of B⁢(H) that are ”rich” in projections) over an infinite-dimensional H, whose lattices of projections are in fact modular.
• •
  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 {0} and a one-dimensional subspace, and every closed subspace other than {0} contains a one-dimensional subspace. This means that the lattice of projections in B⁢(H) is an atomic lattice and its atoms are precisely the projections on one-dimensional subspaces.
  Moreover, every closed subspace of H is the closure of the span of its one-dimensional subspaces. Thus, the lattice of projections in B⁢(H) is an atomistic lattice.
• •
  We shall see further ahead in this entry, when we discuss orthogonal projections, that the lattice of projections in B⁢(H) is an orthomodular lattice.

3 Commuting and Orthogonal Projections

When two projections P,Q commute, the projections P∧Q and P∨Q can be described algebraically in a very . We shall see at the end of this section that P and Q commute precisely when its corresponding subspaces Ran⁢(P) and Ran⁢(Q) are ”perpendicular”.

Theorem 2 - Let P,Q be commuting projections (i.e. P⁢Q=Q⁢P), then

• •
  P∧Q=P⁢Q
• •
  P∨Q=P+Q-P⁢Q
• •
  Ran⁢(P)∨Ran⁢(Q)=Ran⁢(P)+Ran⁢(Q). In particular, Ran⁢(P)+Ran⁢(Q) is closed.

Two projections P,Q are said to be orthogonal if P≤Q⟂. This is equivalent to say that its corresponding subspaces are orthogonal (Ran⁢(P) lies in the orthogonal complement of Ran⁢(Q)).

Corollary 1 - Two projections P,Q are orthogonal if and only if P⁢Q=0. When this is so, then P∨Q=P+Q.

Corollary 2 - Let P,Q be projections in B⁢(H) such that P≤Q. Then Q-P is the projection onto Ran⁢(Q)∩Ran⁢(P)⟂.

We can now see that P,Q commute if and only if Ran⁢(P) and Ran⁢(Q) 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, P and Q commute if and only if the subspaces Ran⁢(P)∩(Ran⁢(P)∩Ran⁢(Q))⟂ and Ran⁢(Q)∩(Ran⁢(P)∩Ran⁢(Q))⟂ are orthogonal.

This can be proved using all the above results: The two subspaces are orthogonal iff

and P⁢Q=P∧Q iff

We can now also see that the lattice of projections is orthomodular: Suppose P≤Q. 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 Λ be a poset. A net of projections {Pα}α∈Λ is said to be increasing if α≤β⟹Pα≤Pβ. Decreasing nets are defined similarly.

Theorem 3 - Let {Pα} be an increasing net of projections. Thenlimα⁡Pα⁢x=⋁αPα⁢x for every x∈H.

Similarly for decreasing nets of projections,

Theorem 4 - Let {Pα} be a decreasing net of projections. Thenlimα⁡Pα⁢x=⋀αPα⁢x for every x∈H.

In other words, Pα converges to ⋀αPα in the strong operator topology.

Theorem 5 - Let Λ be a set and {Pα}α∈Λ be a family of pairwise orthogonal projections. Then ∑Pα is summable and ∑Pα⁢x=⋁αPα⁢x for all x∈H.