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.

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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”.

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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.

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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 $\mathrm{Ran}(Q)$).

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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 $Q-P$ is the projection onto $\mathrm{Ran}(Q)\cap \mathrm{Ran}{(P)}^{⟂}$.

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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 increasingif $\alpha \le \beta ⟹P_{\alpha }\le P_{\beta }$. Decreasing nets aredefined similarly.

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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 operatortopology.

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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 operatortopology.

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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 $x\in H$.