lattice of projections

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

⟷

Ran

⁢

(

P

)

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:

P

≤

Q

⟺

Ran

⁢

(

P

)

⊆

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

≤

s

⁢

a

, and whose definition we recall now

P

≤

s

⁢

a

Q

⟺

Q

-

P

⁢

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

:

•

Ran

⁢

(

P

)

⊆

Ran

⁢

(

Q

)

(i.e.

P

≤

Q

)

•

Q

⁢

P

=

P

•

P

⁢

Q

=

P

•

∥

P

⁢

x

∥

≤

∥

Q

⁢

x

∥

for all

x

∈

H

•

P

≤

s

⁢

a

Q

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

{

Y

α

}

of closed subspaces in

H

possesses an

infimum

⋀

Y

α

and a

supremum

⋁

Y

α

, which are, respectively, the intersection of all

Y

α

and the closure of the subspace generated by all

Y

α

. There is, of course, a correspondent in terms of projections: every family

{

P

α

}

of projections has an infimum

⋀

P

α

and a supremum

⋁

P

α

, which are, respectively, the projection onto the intersection of all

Ran

⁢

(

P

α

)

and the projection onto the closure of the subspace generated by all

Ran

⁢

(

P

α

)

.

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

⊕

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

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

∧

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

0

=

(

P

-

P

∧

Q

)

⁢

(

Q

-

P

∧

Q

)

=

P

⁢

Q

-

P

∧

Q

and

P

⁢

Q

=

P

∧

Q

iff

P

⁢

Q

=

P

∧

Q

=

(

P

∧

Q

)

*

=

(

P

⁢

Q

)

*

=

Q

⁢

P

We can now also see that the lattice of projections is orthomodular: Suppose

P

≤

Q

. Then, using the above results,

P

∨

(

Q

∧

P

⟂

)

=

P

∨

(

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

Λ

be a poset. A net of projections

{

P

α

}

α

∈

Λ

is said to be increasing
if

α

≤

β

⟹

P

α

≤

P

β

. Decreasing nets are
defined similarly.

Theorem 3 -

Let

{

P

α

}

be an increasing net of projections. Then

lim

α

⁡

P

α

⁢

x

=

⋁

α

P

α

⁢

x

for every

x

∈

H

.

In other words,

P

α

converges

to

⋁

α

P

α

in the

strong operator
topology

.

Similarly for decreasing nets of projections,

Theorem 4 -

Let

{

P

α

}

be a decreasing net of projections. Then

lim

α

⁡

P

α

⁢

x

=

⋀

α

P

α

⁢

x

for every

x

∈

H

.

In other words,

P

α

converges to

⋀

α

P

α

in the strong operator
topology.

Theorem 5 -

Let

Λ

be a set and

{

P

α

}

α

∈

Λ

be a family of pairwise orthogonal projections. Then

∑

P

α

is summable and

∑

P

α

⁢

x

=

⋁

α

P

α

⁢

x

for all

x

∈

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

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