PlanetMath: representations of compact groups are equivalent to unitary representations

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representations of compact groups are equivalent to unitary representations

(Theorem)

Theorem - Let be a compact topological group. If $(\pi, V)$ is a finite-dimensional representation of in a normed vector space , then $\pi$ is equivalent to a unitary representation.

$\,$

Proof: Let $\langle \cdot, \cdot \rangle$ denote an inner product in . Define a new inner product in the vector space by

$\displaystyle \langle v, w \rangle_1 := \int_G \big\langle \pi(s)v, \pi(s)w \big\rangle \;d\mu(s)$

where $\mu$ is a Haar measure in

. It is easy to see that $\langle \cdot, \cdot \rangle_1$ defines indeed an inner product, noting that $\langle \pi(\cdot)v, \pi(\cdot)w \rangle$ is a continuous function in

Now we claim that, for every $s \in G$ , $\pi(s)$ is a unitary operator for this new inner product. This is true since

$\displaystyle \big\langle \pi(t)v, \pi(t)w \big\rangle_1$	$\displaystyle =$	$\displaystyle \int_G \big\langle \pi(s)\pi(t)v, \pi(s)\pi(t)w \big\rangle \;d\mu(s)$
	$\displaystyle =$	$\displaystyle \int_G \langle \pi(st)v, \pi(st)w \rangle \;d\mu(s)$
	$\displaystyle =$	$\displaystyle \int_G \langle \pi(s)v, \pi(s)w \rangle \;d\mu(s)$
	$\displaystyle =$	$\displaystyle \langle v, w \rangle_1$

Denote by the space endowed the inner product $\langle \cdot, \cdot \rangle_1$ . As we have seen, $(\pi, V')$ is a unitary representation of in . Of course, $(\pi, V')$ and $(\pi, V)$ are equivalent representations, since

$\displaystyle \pi=\mathrm{id}^{-1}\,\pi\,\mathrm{id}$

where $\mathrm{id}:V \longrightarrow V'$ is the identity mapping. Thus, $(\pi, V)$ is equivalent to a unitary representation. $\square$

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Keywords:

compact group, unitary representation

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Cross-references: identity mapping, unitary operator, continuous function, easy to see, Haar measure, vector space, inner product, unitary representation, normed vector space, topological group, compact

This is version 6 of representations of compact groups are equivalent to unitary representations, born on 2008-05-02, modified 2008-05-03.
Object id is 10563, canonical name is FiniteDimensionalRepresentationOfACompactGroupIsEquivalentToAUnitaryRepresentation.
Accessed 102 times total.

Classification:

AMS MSC:	22A25 (Topological groups, Lie groups :: Topological and differentiable algebraic systems :: Representations of general topological groups and semigroups)
	22C05 (Topological groups, Lie groups :: Compact groups)
	43A65 (Abstract harmonic analysis :: Representations of groups, semigroups, etc.)

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