criterion for a Banach *-algebra representation to be irreducible

Theorem - Let 𝒜 be a Banach *-algebra, H an Hilbert spaceMathworldPlanetmath and I the identity operator in H. A representation ( π:𝒜H is topologically irreducible if and only if π(𝒜)=I, i.e. if and only if the commutant of π(𝒜) consists of scalar multiples of the identity operator.

Proof : ()

As π(𝒜) is selfadjoint, π(𝒜) is a von Neumann algebraMathworldPlanetmathPlanetmathPlanetmath.

Suppose π(𝒜)I. Then the dimensionMathworldPlanetmathPlanetmathPlanetmathPlanetmath of π(𝒜) is greater than one.

It is known that von Neumann algebras of dimension greater than one contain non-trivial projections, so there is a projection Pπ(𝒜) such that P0 and PI.

As Pπ(𝒜), P commutes with every operator Tπ(𝒜), that is PT=TP.

Thus RanP is an invariant subspace of every Tπ(𝒜). Therefore π is not an irreducible representation.


Conversely, suppose that π is not an irreducible representation. There exists a closed π(𝒜)-invariant subspace different from {0} and H.

Let P be the projection onto that closed invariant subspace.

Invariance can be expressed as: π(a)P=Pπ(a)P for every a𝒜. It follows that


for every a𝒜.

We conclude that P commutes with every element of π(𝒜), i.e. Pπ(𝒜).

Thus π(𝒜)I

Title criterion for a Banach *-algebra representation to be irreduciblePlanetmathPlanetmath
Canonical name CriterionForABanachalgebraRepresentationToBeIrreducible
Date of creation 2013-03-22 17:27:43
Last modified on 2013-03-22 17:27:43
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 9
Author asteroid (17536)
Entry type Theorem
Classification msc 46K10