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irreducible unitary representations of compact groups are finite-dimensional (Theorem)

Theorem - If $\pi \in rep(G, H)$ is a unitary representation of a compact topological group $G$ in a Hilbert space $H$ , then $\pi$ has a finite-dimensional subrepresentation.

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Corollary 1 - If $\pi$ is irreducible, then $H$ must be finite-dimensional.

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Corollary 2 - $\pi$ has an irreducible subrepresentation.




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See Also: unitary representation

Other names:  unitary representation of a compact group has a finite-dimensional subrepresentation
Also defines:  unitary representation of compact group has an irreducible subrepresentation, unitary group of a complex Hilbert space
Keywords:  nitary representation of compact group, irreducible subrepresentation, unitary group of a complex Hilbert space
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Cross-references: finite-dimensional, Hilbert space, topological group, compact, unitary representation, theorem

This is version 10 of irreducible unitary representations of compact groups are finite-dimensional, born on 2008-05-07, modified 2008-11-08.
Object id is 10569, canonical name is IrreducibleUnitaryRepresentationsOfCompactGroupsAreFiniteDimensional.
Accessed 1448 times total.

Classification:
AMS MSC22A25 (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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