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

Denote by β„‹ an arbitrary Hilbert spaceMathworldPlanetmath. To fix notation let π’°βŠ‚β„’β’(β„‹) be a Cβˆ— subalgebra of ℒ⁒(β„‹). We then define the commutatorPlanetmathPlanetmath of 𝒰 by

𝒰′ :={Tβˆˆβ„’β’(β„‹):T⁒U=U⁒Tβ’βˆ€Uβˆˆπ’°}

Note that 𝒰′ is closed with regard to the weak topology (see this entry ( So 𝒰′ is always a von Neumann algebraMathworldPlanetmathPlanetmathPlanetmath.

As an immediate consequence of Schur’s Lemma for group representationsMathworldPlanetmathPlanetmath on a Hilbert space we obtain the following result.

Lemma. Let 𝒰 be a βˆ—-algebra and let Ο€ be a βˆ—-representation of 𝒰 on the Hilbert space β„‹. Then Ο€ is topologically irreducible iff π⁒(𝒰)β€²=ℂ⁒I.

We can now prove the result.

Theorem. Let 𝒰 be a Cβˆ— algebra. Assume the βˆ—-representation Ο€ of 𝒰 on the Hilbert space β„‹ is topologically irreducible. Then Ο€ is algebraically irreduciblePlanetmathPlanetmath.


By the Lemma it follows that π⁒(𝒰)β€²=ℂ⁒I. Hence π⁒(𝒰)β€²β€²=ℒ⁒(β„‹). By the double commutant theorem ( every operator in ℒ⁒(β„‹)1 (the unit ballPlanetmathPlanetmath in the set of bounded operatorsMathworldPlanetmathPlanetmath ℒ⁒(β„‹)) belongs to the strong operator closurePlanetmathPlanetmath 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,yβˆˆβ„‹,xβ‰ 0 an element Tβˆˆπ’° such that π⁒(T)⁒x=y holds. Indeed, it is enough to consider the case βˆ₯xβˆ₯=βˆ₯yβˆ₯=1.

Now construct the rank one approximation T~1:=yβŠ—x (⇔T~1⁒z=⟨x,z⟩⁒y,zβˆˆβ„‹β‡’T~1⁒x=βˆ₯xβˆ₯⁒y=y) with a corresponding T1βˆˆπ’°,π⁒(T1)βˆˆΟ€β’(𝒰)1, so that βˆ₯y-π⁒(T1)⁒xβˆ₯=βˆ₯T~1⁒x-π⁒(T1)⁒xβˆ₯≀12.

Approximate further T~2:=(y-π⁒(T1)⁒x)βŠ—x∈12⁒ℒ⁒(β„‹)1 and choose π⁒(T2)∈12⁒π⁒(𝒰)1 with βˆ₯y-π⁒(T1)⁒x-π⁒(T2)⁒xβˆ₯=βˆ₯T~2⁒x-π⁒(T2)⁒xβˆ₯≀122.

Proceed by induction with T~n:=(y-βˆ‘j=1n-1π⁒(Tj)⁒x)βŠ—x∈2-j⁒ℒ⁒(β„‹)1. Choose π⁒(Tn)∈2-n⁒π⁒(𝒰)1 with βˆ₯y-βˆ‘j=1nπ⁒(Tj)⁒xβˆ₯=βˆ₯T~n⁒x-π⁒(Tn)⁒xβˆ₯≀2-n. Then we have π⁒(T):=βˆ‘j=1nπ⁒(Tn) in 𝒰 and π⁒(T)⁒x=y which completesPlanetmathPlanetmathPlanetmath 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