# representations of compact groups are equivalent to unitary representations

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representation
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\PMlinkescapephraseequivalent^{}

Theorem - Let $G$ be a compact topological group. If $(\pi ,V)$ is a finite-dimensional representation (http://planetmath.org/TopologicalGroupRepresentation) of $G$ in a normed vector space^{} $V$, then $\pi $ is equivalent (http://planetmath.org/TopologicalGroupRepresentation) to a unitary representation^{}.

$$

*Proof:* Let $\u27e8\cdot ,\cdot \u27e9$ denote an inner product^{} in $V$. Define a new inner product in the vector space^{} $V$ by

${\u27e8v,w\u27e9}_{1}:={\displaystyle {\int}_{G}}\u27e8\pi (s)v,\pi (s)w\u27e9\mathit{d}\mu (s)$ |

where $\mu $ is a Haar measure in $G$. It is easy to see that ${\u27e8\cdot ,\cdot \u27e9}_{1}$ defines indeed an inner product, noting that $\u27e8\pi (\cdot )v,\pi (\cdot )w\u27e9$ is a continuous function^{} in $G$.

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

${\u27e8\pi (t)v,\pi (t)w\u27e9}_{1}$ | $=$ | ${\int}_{G}}\u27e8\pi (s)\pi (t)v,\pi (s)\pi (t)w\u27e9\mathit{d}\mu (s)$ | ||

$=$ | ${\int}_{G}}\u27e8\pi (st)v,\pi (st)w\u27e9\mathit{d}\mu (s)$ | |||

$=$ | ${\int}_{G}}\u27e8\pi (s)v,\pi (s)w\u27e9\mathit{d}\mu (s)$ | |||

$=$ | ${\u27e8v,w\u27e9}_{1}$ |

Denote by ${V}^{\prime}$ the space $V$ endowed the inner product ${\u27e8\cdot ,\cdot \u27e9}_{1}$. As we have seen, $(\pi ,{V}^{\prime})$ is a unitary representation of $G$ in ${V}^{\prime}$. Of course, $(\pi ,{V}^{\prime})$ and $(\pi ,V)$ are equivalent representations, since

$\pi ={\mathrm{id}}^{-1}\pi \mathrm{id}$ |

where $\mathrm{id}:V\u27f6{V}^{\prime}$ is the identity mapping. Thus, $(\pi ,V)$ is equivalent to a unitary representation. $\mathrm{\square}$

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 |