Gelfand-Naimark-Segal construction

Gelfand-Naimark-Segal construction

1

GNS Construction

The

Gelfand-Naimark-Segal construction

(or GNS construction) is a fundamental idea in the of . It provides a procedure to construct and study

representations

of

C

*

-

algebras

(

http://planetmath.org/CAlgebra

) and is the first step on the proof of the

Gelfand-Naimark representation theorem

, which that every

C

*

-algebra is

isometrically isomorphic

to a closed *-

subalgebra

of

B

⁢

(

H

)

, the algebra of

bounded operators

on a

Hilbert space

H

.

There are

generalizations

of this construction for Banach *-algebras with an

approximate unit

, and some of the results stated here are in fact valid for this kind of algebras, but we will restrict our attention to the

C

*

case.

2

Representations associated with positive linear functionals

Let

𝒜

be a

C

*

-algebra and

ϕ

a

positive linear functional

in

𝒜

.

We are going to construct a representation

π

ϕ

of

𝒜

and for that we need to construct a suitable Hilbert space.

Let us endow

𝒜

with a

semi-inner product

defined by

⟨

x

,

y

⟩

ϕ

:=

ϕ

⁢

(

y

*

⁢

x

)

. Now we define the set

N

ϕ

:=

{

x

∈

𝒜

:

⟨

x

,

x

⟩

ϕ

=

0

}

It is easily seen that

N

ϕ

is a closed

left ideal

(

http://planetmath.org/Ideal

) in

𝒜

(using the

Cauchy-Schwarz inequality

, which is valid in semi-inner product spaces), so that

⟨

⋅

,

⋅

⟩

ϕ

induces a well defined

inner product

on the

quotient

(

http://planetmath.org/QuotientModule

)

𝒜

/

N

ϕ

. The completion of

𝒜

/

N

ϕ

is then an Hilbert space, which we will be denoted by

H

ϕ

.

We will now define a representation of

𝒜

on

H

ϕ

by left

multiplication

. For every

a

∈

𝒜

let

π

ϕ

⁢

(

a

)

be the

operator

of left multiplication by

a

on

𝒜

/

N

ϕ

, i.e.

π

⁢

(

a

)

⁢

(

x

+

N

ϕ

)

:=

a

⁢

x

+

N

ϕ

Theorem

1 -

The

function

π

ϕ

⁢

(

a

)

:

𝒜

/

N

ϕ

⟶

𝒜

/

N

ϕ

defined above is linear and

bounded

(

http://planetmath.org/BoundedOperator

), with

∥

π

ϕ

⁢

(

a

)

∥

≤

∥

a

∥

.

Being bounded, the operator

π

ϕ

⁢

(

a

)

extends uniquely to a bounded operator on

H

ϕ

, which we denote by the same symbol,

π

ϕ

⁢

(

a

)

.

Let

B

⁢

(

H

ϕ

)

be the algebra of bounded operators on

H

ϕ

.

Theorem 2 -

The function

π

ϕ

:

𝒜

⟶

B

⁢

(

H

ϕ

)

defined by

a

↦

π

ϕ

⁢

(

a

)

is a

C

*

-algebra representation of

𝒜

.

This representation is called the

GNS representation

associated to

ϕ

.

3

Cyclic vectors and GNS pairs

Suppose

𝒜

had an

identity element

e

. In this case it is easily seen that there exists a

cyclic vector

ξ

ϕ

∈

H

ϕ

, i.e. a vector

ξ

ϕ

such that

π

ϕ

⁢

(

𝒜

)

⁢

ξ

ϕ

is dense (

http://planetmath.org/Dense

) in

H

ϕ

. This cyclic vector can just be chosen as

e

+

N

ϕ

.

Moreoever, this cyclic vector

ξ

ϕ

:=

e

+

N

ϕ

is such that

ϕ

⁢

(

a

)

=

⟨

π

ϕ

⁢

(

a

)

⁢

ξ

ϕ

,

ξ

ϕ

⟩

ϕ

for every

a

∈

𝒜

.

Thus, in this case the representation

π

ϕ

is cyclic (

http://planetmath.org/BanachAlgebraRepresentation

) and

ϕ

is a vector state of

𝒜

. The result is still valid for general

C

*

-algebras:

Theorem 3 -

Let

π

ϕ

be the representation of

𝒜

defined previously. Then there exists a vector

ξ

ϕ

∈

H

ϕ

such that

•

π

ϕ

⁢

(

𝒜

)

⁢

ξ

ϕ

is dense in

H

ϕ

, i.e.

π

ϕ

is cyclic,

•

ϕ

⁢

(

a

)

=

⟨

π

ϕ

⁢

(

a

)

⁢

ξ

ϕ

,

ξ

ϕ

⟩

ϕ

for every

a

∈

𝒜

, i.e.

ϕ

is a vector state.

Any pair

(

π

,

ξ

)

, where

π

is a representation of

𝒜

on a Hilbert space

H

and

ξ

∈

H

, satisfying the above conditions for

ϕ

:

•

π

⁢

(

𝒜

)

⁢

ξ

is dense in

H

,

•

ϕ

⁢

(

a

)

=

⟨

π

⁢

(

a

)

⁢

ξ

,

ξ

⟩

for every

a

∈

𝒜

is called a

GNS pair

for

ϕ

.

Theorem 4 -

All GNS pairs for

ϕ

are (in the sense that the corresponding representations are

unitarily equivalent

).

4

Irreducible representations

We know that are ”plenty” of states on

C

*

-algebra (see this entry (

http://planetmath.org/PropertiesOfStates

)), and so we have assured the

existence

of many (cyclic) representations. An interesting fact is that this representations associated to states are irreducible (

http://planetmath.org/BanachAlgebraRepresentation

) exactly when the state is a

pure state

:

Theorem 5 -

Let

ϕ

be a state on

𝒜

. Then the representation

π

ϕ

is irreducible if and only if

ϕ

is a pure state.

The fact that there are ”plenty” of pure states in a

C

*

-algebra allows one to assure the existence of

irreducible representations

that preserve the norm of a given

element

in

𝒜

.

Theorem 6 -

Let

𝒜

be a

C

*

-algebra. For every element

a

there exists an irreducible representation

π

of

𝒜

such that

∥

π

⁢

(

a

)

∥

=

∥

a

∥

.

This last theorem is a fundamental step in the proof of the Gelfand-Naimark representation theorem.

Title

Gelfand-Naimark-Segal construction

Canonical name

GelfandNaimarkSegalConstruction

Date of creation

2013-03-22 17:47:40

Last modified on

2013-03-22 17:47:40

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

10

Author

asteroid (17536)

Entry type

Feature

Classification

msc 46L30

Classification

msc 46L05

Synonym

GNS construction

Related topic

CAlgebra

Related topic

CAlgebra3

Related topic

RepresentationOfAC_cG_dTopologicalAlgebra

Defines

GNS pair

Defines

GNS representation

Defines

pure states and irreducible representations

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