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Linear representations

  1. Nov 17, 2008 #1


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    Let H be a separable Hilbert space. What are the continuous
    linear representations of S^1 on H?

    I read in an article this is defined as in the finite-dim case.
    Why is this so?

  2. jcsd
  3. Nov 17, 2008 #2


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    Well S^1 is compact, so every continuous linear representation of S^1 on H is unitary (re-norm H if necessary) and decomposes into a direct sum of irreducible representations. And since S^1 is abelian, its irreducible representations are nothing other than its characters, i.e. they're one-dimensional.

    Is this what you're asking?
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