representations of compact groups are equivalent to unitary representations


representation \PMlinkescapephraserepresentations \PMlinkescapephraseequivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath

Theorem - Let G be a compact topological group. If (π,V) is a finite-dimensional representation ( of G in a normed vector spacePlanetmathPlanetmath V, then π is equivalent ( to a unitary representationMathworldPlanetmath.

Proof: Let , denote an inner productMathworldPlanetmath in V. Define a new inner product in the vector spaceMathworldPlanetmath V by


where μ is a Haar measure in G. It is easy to see that ,1 defines indeed an inner product, noting that π()v,π()w is a continuous functionPlanetmathPlanetmath in G.

Now we claim that, for every sG, π(s) is a unitary operator for this new inner product. This is true since

π(t)v,π(t)w1 = Gπ(s)π(t)v,π(s)π(t)w𝑑μ(s)
= Gπ(st)v,π(st)w𝑑μ(s)
= Gπ(s)v,π(s)w𝑑μ(s)
= v,w1

Denote by V the space V endowed the inner product ,1. As we have seen, (π,V) is a unitary representation of G in V. Of course, (π,V) and (π,V) are equivalent representations, since


where id:VV is the identity mapping. Thus, (π,V) is equivalent to a unitary representation.

Title representations of compact groups are equivalent to unitary representations
Canonical name RepresentationsOfCompactGroupsAreEquivalentToUnitaryRepresentations
Date of creation 2013-03-22 18:02:28
Last modified on 2013-03-22 18:02:28
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 9
Author asteroid (17536)
Entry type Theorem
Classification msc 43A65
Classification msc 22A25
Classification msc 22C05