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) selfadjoint 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
$$P\u27f7\mathrm{Ran}(P)$$ 
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:
$$P\le Q\u27fa\mathrm{Ran}(P)\subseteq \mathrm{Ran}(Q)$$ 
But since projections are selfadjoint operators (in fact they are positive operators, as $P={P}^{*}P$), they inherit the natural partial ordering of selfadjoint operators (http://planetmath.org/OrderingOfSelfAdjoints), which we denote by ${\le}_{sa}$, and whose definition we recall now
$$P{\le}_{sa}Q\u27faQP\text{is a positive operator}$$ 
As the following theorem shows, these two orderings coincide. Thus, we shall not make any more distinctions of notation between them.
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Theorem 1  Let $P,Q$ be projections in $B(H)$. The following conditions are equivalent^{}:

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

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$QP=P$

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$PQ=P$

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

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$P{\le}_{sa}Q$
$$
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

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The lattice of projections in $B(H)$ is never distributive^{} (http://planetmath.org/DistributiveLattice) (unless $H$ is onedimensional).

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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 infinitedimensional $H$, whose lattices of projections are in fact modular.

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Projections on onedimensional 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 onedimensional subspace, and every closed subspace other than $\{0\}$ contains a onedimensional subspace. This means that the lattice of projections in $B(H)$ is an atomic lattice and its atoms are precisely the projections on onedimensional subspaces.
Moreover, every closed subspace of $H$ is the closure of the span of its onedimensional subspaces. Thus, the lattice of projections in $B(H)$ is an atomistic lattice.

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In Hilbert spaces every closed subspace $Z$ is topologically complemented by its orthogonal complement^{} ($H=Z\oplus {Z}^{\u27c2}$), 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 $IP$ (onto $\mathrm{Ran}{(P)}^{\u27c2}$).

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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^{}”.
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Theorem 2  Let $P,Q$ be commuting projections (i.e. $PQ=QP$), then

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$P\wedge Q=PQ$

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$P\vee Q=P+QPQ$

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$\mathrm{Ran}(P)\vee \mathrm{Ran}(Q)=\mathrm{Ran}(P)+\mathrm{Ran}(Q)$. In particular, $\mathrm{Ran}(P)+\mathrm{Ran}(Q)$ is closed.
$$
Two projections $P,Q$ are said to be orthogonal^{} if $P\le {Q}^{\u27c2}$. This is equivalent to say that its corresponding subspaces are orthogonal ($\mathrm{Ran}(P)$ lies in the orthogonal complement of $\mathrm{Ran}(Q)$).
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Corollary 1  Two projections $P,Q$ are orthogonal if and only if $PQ=0$. When this is so, then $P\vee Q=P+Q$.
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Corollary 2  Let $P,Q$ be projections in $B(H)$ such that $P\le Q$. Then $QP$ is the projection onto $\mathrm{Ran}(Q)\cap \mathrm{Ran}{(P)}^{\u27c2}$.
$$
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 $\mathrm{Ran}(P)\cap {(\mathrm{Ran}(P)\cap \mathrm{Ran}(Q))}^{\u27c2}$ and $\mathrm{Ran}(Q)\cap {(\mathrm{Ran}(P)\cap \mathrm{Ran}(Q))}^{\u27c2}$ are orthogonal.
This can be proved using all the above results: The two subspaces are orthogonal iff
$$0=(PP\wedge Q)(QP\wedge Q)=PQP\wedge Q$$ 
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,
$$P\vee (Q\wedge {P}^{\u27c2})=P\vee (QP)=P+(QP)P(QP)=Q$$ 
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 $\alpha \le \beta \u27f9{P}_{\alpha}\le {P}_{\beta}$. Decreasing nets are defined similarly.
$$
Theorem 3  Let $\{{P}_{\alpha}\}$ be an increasing net of projections. Then ${lim}_{\alpha}{P}_{\alpha}x={\bigvee}_{\alpha}{P}_{\alpha}x$ for every $x\in H$.
In other words, ${P}_{\alpha}$ converges^{} to ${\bigvee}_{\alpha}{P}_{\alpha}$ in the strong operator topology.
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Similarly for decreasing nets of projections,
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Theorem 4  Let $\{{P}_{\alpha}\}$ be a decreasing net of projections. Then ${lim}_{\alpha}{P}_{\alpha}x={\bigwedge}_{\alpha}{P}_{\alpha}x$ for every $x\in H$.
In other words, ${P}_{\alpha}$ converges to ${\bigwedge}_{\alpha}{P}_{\alpha}$ in the strong operator topology.
$$
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  20130322 17:53:29 
Last modified on  20130322 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 