projections as noncommutative characteristic functions

projections as noncommutative characteristic functions

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similarities

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point

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points

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positive

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observations

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observation

In this entry we try to exhibit the profound similarities there are between

projections in Hilbert spaces

and

characteristic functions

in a

measure space

. In fact, in the general framing of viewing

von Neumann algebras

as noncommutative measure spaces,

projections

are the noncommutative analog of characteristic functions.

Note -

By a projection we always an orthogonal projection.

Let us some notation first:

•

H

denotes a

Hilbert space

and

B

⁢

(

H

)

its algebra of

bounded operators

.

•

(

X

,

𝔅

,

μ

)

denotes a measure space.

•

χ

A

denotes the characteristic function of the

measurable set

A

⊂

X

.

Recall that a projection in

B

⁢

(

H

)

is a bounded operator

P

such that

P

*

⁢

P

=

P

. Let us now point out what projections and characteristic functions have in common. Note that, although we have written our observations in separate points (making it easier to read), they are all closely related.

0.1

Basic Facts

•

Just like projections, characteristic functions

satisfy

:

χ

A

2

=

χ

A

. Thus, both characteristic functions and projections are idempotents.

•

Characteristic functions are

functions

(meaning they take only positive or zero values), just like projections are positive operators.

•

Characteristic functions take only the values

0

and

1

. The

spectrum

of a projection is

contained

in

{

0

,

1

}

. Notice that the spectrum of a

normal operator

consists precisely of the values of the complex valued function associated with it (using the

Gelfand-Naimark theorem

and/or the

continuous functional calculus

).

•

A

partial ordering

can be defined on the set of characteristic functions by saying

χ

A

≤

χ

B

⟺

A

⊆

B

⁢

, or equivalently,

⁢

χ

A

≤

χ

B

⟺

χ

B

-

χ

A

⁢

is a positive function

.

Analogously, a partial ordering can be defined on the set of projections by saying

P

≤

Q

⟺

Ran

⁢

(

P

)

⊆

Ran

⁢

(

Q

)

⁢

, or equivalently,

⁢

P

≤

Q

⟺

Q

-

P

⁢

is a positive operator

.

where

Ran

⁢

(

P

)

denotes the range of the

operator

P

.

0.2

Projections of

L

∞

⁢

(

X

,

μ

)

The above observations could all be easily derived from the general fact we describe next:

•

Consider the

Banach algebra

L

∞

⁢

(

X

,

μ

)

. Functions

f

∈

L

∞

⁢

(

X

,

μ

)

can be seen as

multiplication operators

in the Hilbert space

L

2

⁢

(

X

,

μ

)

. Thus,

L

∞

⁢

(

X

,

μ

)

can be seen as a closed

subalgebra

of

B

⁢

(

L

2

⁢

(

X

,

μ

)

)

(it is in fact a von Neumann algebra).

Characteristic functions in

L

∞

⁢

(

X

,

μ

)

are exactly the projections of this subalgebra.

0.3

Measure Theory and the Spectral Theorem

The next key observation explores the similarities between some facts about

measure theory

and the

spectral theorem

of

self-adjoint

(or normal) operators.

It is a well known fact from measure theory that a

continuous function

f

:

X

⟶

ℝ

can be approximated by

linear combinations

of characteristic functions. With some additional effort it can be seen that, in fact, each continuous function

f

is a (vector valued)

integral

of characteristic functions

f

=

∫

X

f

⁢

𝑑

χ

where

χ

is the

vector measure

of characteristic functions

χ

⁢

(

A

)

:=

χ

A

.

An analogous phenomenon in the

spectral theory

of normal operators. Notice (as pointed earlier) that the

C

*

-algebra (

http://planetmath.org/CAlgebra

) theory allows one to see a normal operator as a continuous function. With this in mind, the spectral theorem of normal operators can be seen as an analog of the previous

measure

theoretic construction. Recall that the spectral theorem that a normal operator

N

can be approximated by linear combinations of projections and can, in fact, be given by a (vector valued) integral of projections:

N

=

∫

σ

⁢

(

N

)

λ

⁢

𝑑

P

⁢

(

λ

)

where

σ

⁢

(

N

)

denotes the spectrum of

N

and

P

is the

projection valued measure

associated with

N

.

Title

projections as noncommutative characteristic functions

Canonical name

ProjectionsAsNoncommutativeCharacteristicFunctions

Date of creation

2013-03-22 17:54:43

Last modified on

2013-03-22 17:54:43

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

11

Author

asteroid (17536)

Entry type

Feature

Classification

msc 46C07

Classification

msc 46L10

Classification

msc 46L51

Classification

msc 46C05

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