lattice of projections

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

Recall that a projection $P$ in $B(H)$ is a bounded (http://planetmath.org/BoundedOperator) self-adjoint operator satisfying $P^2=P$.

The set of projections in $B(H)$, 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

$$P⟷\mathrm{Ran}(P)$$

where $P$ is a projection and $\mathrm{Ran}(P)$ denotes the range of $P$.

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

$$P\le Q⟺\mathrm{Ran}(P)\subseteq \mathrm{Ran}(Q)$$

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 ${\le }_{sa}$, and whose definition we recall now

$$P{\le }_{sa}Q⟺Q-P\text{is a positive operator}$$

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 equivalent:

• $\mathrm{Ran}(P)\subseteq \mathrm{Ran}(Q)$ (i.e. $P\le Q$)

• $QP=P$

• $PQ=P$

• $\parallel Px\parallel \le \parallel Qx\parallel$ for all $x\in H$

• $P{\le }_{sa}Q$

Two closed subspaces $Y,Z$ in $H$ have a greatest lower bound $Y\wedge Z$ and a least upper bound $Y\vee Z$. Specifically, $Y\wedge Z$ is precisely the intersection $Y\cap Z$ and $Y\vee Z$ is precisely the closure of the subspace generated by $Y$ and $Z$. Hence, if $P,Q$ are projections in $B(H)$ then $P\wedge Q$ is the projection onto $\mathrm{Ran}(P)\cap \mathrm{Ran}(Q)$ and $P\vee Q$ is the projection onto the closure of $\mathrm{Ran}(P)+\mathrm{Ran}(Q)$.

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 $\{Y_{\alpha }\}$ of closed subspaces in $H$ possesses an infimum $\bigwedge Y_{\alpha }$ and a supremum $\bigvee Y_{\alpha }$, which are, respectively, the intersection of all $Y_{\alpha }$ and the closure of the subspace generated by all $Y_{\alpha }$. There is, of course, a correspondent in terms of projections: every family $\{P_{\alpha }\}$ of projections has an infimum $\bigwedge P_{\alpha }$ and a supremum $\bigvee P_{\alpha }$, which are, respectively, the projection onto the intersection of all $\mathrm{Ran}(P_{\alpha })$ and the projection onto the closure of the subspace generated by all $\mathrm{Ran}(P_{\alpha })$.

2 Additional Lattice Features

• The lattice of projections in $B(H)$ is never distributive (http://planetmath.org/DistributiveLattice) (unless $H$ is one-dimensional).

• 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.

• In Hilbert spaces every closed subspace $Z$ is topologically complemented by its orthogonal complement ($H=Z\oplus Z^{⟂}$), and this fact is reflected in the structure of projections. The lattice of projections is then an orthocomplemented lattice, where the orthocomplement of each projection $P$ is the projection $I-P$ (onto $\mathrm{Ran}{(P)}^{⟂}$).

• 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 $P\wedge Q$ and $P\vee Q$ 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 $\mathrm{Ran}(P)$ and $\mathrm{Ran}(Q)$ are ”perpendicular”.

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

• $P\wedge Q=PQ$

• $P\vee Q=P+Q-PQ$

• $\mathrm{Ran}(P)\vee \mathrm{Ran}(Q)=\mathrm{Ran}(P)+\mathrm{Ran}(Q)$. In particular, $\mathrm{Ran}(P)+\mathrm{Ran}(Q)$ is closed.

Two projections $P,Q$ are said to be orthogonal if $P\le Q^{⟂}$. This is equivalent to say that its corresponding subspaces are orthogonal ($\mathrm{Ran}(P)$ lies in the orthogonal complement of $\mathrm{Ran}(Q)$).

Corollary 1 - Two projections $P,Q$ are orthogonal if and only if $PQ=0$. When this is so, then $P\vee Q=P+Q$.

Corollary 2 - Let $P,Q$ be projections in $B(H)$ such that $P\le Q$. Then $Q-P$ is the projection onto $\mathrm{Ran}(Q)\cap \mathrm{Ran}{(P)}^{⟂}$.

We can now see that $P,Q$ commute if and only if $\mathrm{Ran}(P)$ and $\mathrm{Ran}(Q)$ 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 $\mathrm{Ran}(P)\cap {(\mathrm{Ran}(P)\cap \mathrm{Ran}(Q))}^{⟂}$ and $\mathrm{Ran}(Q)\cap {(\mathrm{Ran}(P)\cap \mathrm{Ran}(Q))}^{⟂}$ are orthogonal.

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

$$0=(P-P\wedge Q)(Q-P\wedge Q)=PQ-P\wedge Q$$

and $PQ=P\wedge Q$ iff

$$PQ=P\wedge Q={(P\wedge Q)}^{*}={(PQ)}^{*}=QP$$

We can now also see that the lattice of projections is orthomodular: Suppose $P\le Q$. Then, using the above results,

$$P\vee (Q\wedge P^{⟂})=P\vee (Q-P)=P+(Q-P)-P(Q-P)=Q$$

4 Nets of Projections

In the following we discuss some useful and interesting results about convergence and limits of projections.

Let $\mathrm{\Lambda }$ be a poset. A net of projections ${\{P_{\alpha }\}}_{\alpha \in \mathrm{\Lambda }}$ is said to be increasing if $\alpha \le \beta ⟹P_{\alpha }\le P_{\beta }$. Decreasing nets are defined similarly.

Theorem 3 - Let $\{P_{\alpha }\}$ be an increasing net of projections. Then ${lim}_{\alpha }{P}_{\alpha }x={\bigvee }_{\alpha }{P}_{\alpha }x$ for every $x\in H$.

In other words, $P_{\alpha }$ converges to ${\bigvee }_{\alpha }{P}_{\alpha }$ in the strong operator topology.

Similarly for decreasing nets of projections,

Theorem 4 - Let $\{P_{\alpha }\}$ be a decreasing net of projections. Then ${lim}_{\alpha }{P}_{\alpha }x={\bigwedge }_{\alpha }{P}_{\alpha }x$ for every $x\in H$.

In other words, $P_{\alpha }$ converges to ${\bigwedge }_{\alpha }{P}_{\alpha }$ in the strong operator topology.

Theorem 5 - Let $\mathrm{\Lambda }$ be a set and ${\{P_{\alpha }\}}_{\alpha \in \mathrm{\Lambda }}$ be a family of pairwise orthogonal projections. Then $\sum P_{\alpha }$ is summable and $\sum P_{\alpha }x={\bigvee }_{\alpha }{P}_{\alpha }x$ for all $x\in H$.

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