proof of topologically irreducible representations are algebraically irreducible for C^*-algebras

proof of topologically irreducible representations are algebraically irreducible for

C

*

-algebras

Denote by

ℋ

an arbitrary

Hilbert space

.
To fix notation let

𝒰

⊂

ℒ

⁢

(

ℋ

)

be a

C

∗

subalgebra

of

ℒ

⁢

(

ℋ

)

. We then define the

commutator

of

𝒰

by

𝒰

′

:=

{

T

∈

ℒ

⁢

(

ℋ

)

:

T

⁢

U

=

U

⁢

T

⁢

∀

U

∈

𝒰

}

Note that

𝒰

′

is closed with regard to the

weak topology

(see this entry (

http://planetmath.org/CommutantIsAWeakOperatorClosedSubalgebra

)). So

𝒰

′

is always a

von Neumann algebra

.

As an immediate consequence of Schur’s Lemma for

group representations

on a Hilbert space we obtain the following result.

Lemma.

Let

𝒰

be a

∗

-

algebra

and let

π

be a

∗

-representation of

𝒰

on the Hilbert space

ℋ

. Then

π

is

topologically irreducible

iff

π

⁢

(

𝒰

)

′

=

ℂ

⁢

I

.

We can now prove the result.

Theorem.

Let

𝒰

be a

C

∗

algebra. Assume the

∗

-representation

π

of

𝒰

on the Hilbert space

ℋ

is topologically irreducible. Then

π

is algebraically

irreducible

.

Proof.

By the Lemma it follows that

π

⁢

(

𝒰

)

′

=

ℂ

⁢

I

. Hence

π

⁢

(

𝒰

)

′′

=

ℒ

⁢

(

ℋ

)

. By the

double commutant theorem

(

http://planetmath.org/VonNeumannDoubleCommutantTheorem

) every operator in

ℒ

⁢

(

ℋ

)

1

(the

unit ball

in the set of

bounded operators

ℒ

⁢

(

ℋ

)

) belongs to the strong operator

closure

of

π

⁢

(

𝒰

)

1

(the unit ball in

π

⁢

(

𝒰

)

).

To show the algebraical irreducibility of

π

⁢

(

𝒰

)

it is enough to find for two given vectors

x

,

y

∈

ℋ

,

x

≠

0

an element

T

∈

𝒰

such that

π

⁢

(

T

)

⁢

x

=

y

holds. Indeed, it is enough to consider the case

∥

x

∥

=

∥

y

∥

=

1

.

Now

construct

the rank one approximation

T

~

1

:=

y

⊗

x

(

⇔

T

~

1

⁢

z

=

⟨

x

,

z

⟩

⁢

y

,

z

∈

ℋ

⇒

T

~

1

⁢

x

=

∥

x

∥

⁢

y

=

y

) with a corresponding

T

1

∈

𝒰

,

π

⁢

(

T

1

)

∈

π

⁢

(

𝒰

)

1

, so that

∥

y

-

π

⁢

(

T

1

)

⁢

x

∥

=

∥

T

~

1

⁢

x

-

π

⁢

(

T

1

)

⁢

x

∥

≤

1

2

.

Approximate further

T

~

2

:=

(

y

-

π

⁢

(

T

1

)

⁢

x

)

⊗

x

∈

1

2

⁢

ℒ

⁢

(

ℋ

)

1

and choose

π

⁢

(

T

2

)

∈

1

2

⁢

π

⁢

(

𝒰

)

1

with

∥

y

-

π

⁢

(

T

1

)

⁢

x

-

π

⁢

(

T

2

)

⁢

x

∥

=

∥

T

~

2

⁢

x

-

π

⁢

(

T

2

)

⁢

x

∥

≤

1

2

2

.

Proceed by induction with

T

~

n

:=

(

y

-

∑

j

=

1

n

-

1

π

⁢

(

T

j

)

⁢

x

)

⊗

x

∈

2

-

j

⁢

ℒ

⁢

(

ℋ

)

1

. Choose

π

⁢

(

T

n

)

∈

2

-

n

⁢

π

⁢

(

𝒰

)

1

with

∥

y

-

∑

j

=

1

n

π

⁢

(

T

j

)

⁢

x

∥

=

∥

T

~

n

⁢

x

-

π

⁢

(

T

n

)

⁢

x

∥

≤

2

-

n

. Then we have

π

⁢

(

T

)

:=

∑

j

=

1

n

π

⁢

(

T

n

)

in

𝒰

and

π

⁢

(

T

)

⁢

x

=

y

which

completes

the proof.
∎

Title

proof of topologically irreducible representations are algebraically irreducible for

C

*

-algebras

Canonical name

ProofOfTopologicallyIrreducibleRepresentationsAreAlgebraicallyIrreducibleForCalgebras

Date of creation

2013-03-22 19:04:12

Last modified on

2013-03-22 19:04:12

Owner

karstenb (16623)

Last modified by

karstenb (16623)

Numerical id

8

Author

karstenb (16623)

Entry type

Proof

Classification

msc 46L05

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