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Math Help - unital *-homomorphism

  1. #1
    Member Mauritzvdworm's Avatar
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    unital *-homomorphism

    If we have a unital *-homomorphism \varphi:M_2(\mathbb{C})\rightarrow M_2(\mathbb{C}) show that is must be a *-automorphism.
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  2. #2
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    Opalg's Avatar
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    Quote Originally Posted by Mauritzvdworm View Post
    If we have a unital *-homomorphism \varphi:M_2(\mathbb{C})\rightarrow M_2(\mathbb{C}) show that is must be a *-automorphism.
    The kernel of a *-homomorphism is a two-sided ideal, and there aren't many of those in M_2(\mathbb{C}).
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  3. #3
    Member Mauritzvdworm's Avatar
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    Aha, I see. Since M_2(\mathbb{C}) is simple it only has two closed two sided ideals, 0 and itself.
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