If we have a unital *-homomorphism $\displaystyle \varphi:M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ show that is must be a *-automorphism.
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Originally Posted by Mauritzvdworm If we have a unital *-homomorphism $\displaystyle \varphi:M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ 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 $\displaystyle M_2(\mathbb{C})$.
Aha, I see. Since $\displaystyle M_2(\mathbb{C})$ is simple it only has two closed two sided ideals, 0 and itself.
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