lattice of projections

Let H be a Hilbert spaceMathworldPlanetmath and B(H) the algebra of bounded operatorsMathworldPlanetmathPlanetmath in H. By a projection in B(H) we always an orthogonal projection.

Recall that a projection P in B(H) is a boundedPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ( self-adjoint operator satisfying P2=P.

The set of projections in B(H), although not forming a vector spaceMathworldPlanetmath, has a very rich structureMathworldPlanetmath. In this entry we are going to endow this set with a partial ordering in a that it becomes a complete latticeMathworldPlanetmath. The latticeMathworldPlanetmath structure of the set of projections has profound consequences on the structure of von Neumann algebrasMathworldPlanetmathPlanetmathPlanetmath.

1 The Lattice of Projections

In Hilbert spaces there is a bijectiveMathworldPlanetmathPlanetmath correspondence between closed subspaces and projections (see this entry ( 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 (, which we denote by sa, and whose definition we recall now

PsaQQ-Pis a positive operator

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 equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

  • Ran(P)Ran(Q) (i.e. PQ)

  • QP=P

  • PQ=P

  • PxQx for all xH

  • PsaQ

Two closed subspaces Y,Z in H have a greatest lower boundMathworldPlanetmath YZ and a least upper bound YZ. Specifically, YZ is precisely the intersectionMathworldPlanetmath YZ and YZ is precisely the closureMathworldPlanetmathPlanetmathPlanetmath of the subspaceMathworldPlanetmathPlanetmathPlanetmath generated by Y and Z. Hence, if P,Q are projections in B(H) then PQ is the projection onto Ran(P)Ran(Q) and PQ 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 infimumMathworldPlanetmath Yα and a supremumMathworldPlanetmath 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 modularPlanetmathPlanetmath 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 minimalPlanetmathPlanetmath 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 PQ and PQ can be described algebraically in a very . We shall see at the end of this sectionMathworldPlanetmathPlanetmath that P and Q commute precisely when its corresponding subspaces Ran(P) and Ran(Q) are ”perpendicularMathworldPlanetmathPlanetmathPlanetmath”.

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

  • PQ=PQ

  • PQ=P+Q-PQ

  • Ran(P)Ran(Q)=Ran(P)+Ran(Q). In particular, Ran(P)+Ran(Q) is closed.

Two projections P,Q are said to be orthogonalMathworldPlanetmath if PQ. 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 PQ=0. When this is so, then PQ=P+Q.

Corollary 2 - Let P,Q be projections in B(H) such that PQ. 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 PQ=PQ iff


We can now also see that the lattice of projections is orthomodular: Suppose PQ. 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}αΛ is said to be increasing if αβPαPβ. Decreasing nets are defined similarly.

Theorem 3 - Let {Pα} be an increasing net of projections. Then limαPαx=αPαx for every xH.

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

Similarly for decreasing nets of projections,

Theorem 4 - Let {Pα} be a decreasing net of projections. Then limαPαx=αPαx for every xH.

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

Theorem 5 - Let Λ be a set and {Pα}αΛ be a family of pairwise orthogonal projections. Then Pα is summable and Pαx=αPαx for all xH.

Title lattice of projections
Canonical name LatticeOfProjections
Date of creation 2013-03-22 17:53:29
Last modified on 2013-03-22 17:53:29
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 14
Author asteroid (17536)
Entry type Feature
Classification msc 46C07
Classification msc 46C05
Classification msc 06C15
Classification msc 46L10
Synonym projections in Hilbert spaces
Related topic OrthomodularLattice
Related topic QuantumLogic
Related topic ContinuousGeometry
Defines minimal projection