unital *-homomorphism

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March 21st 2011, 10:11 PM
Mauritzvdworm

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.
March 22nd 2011, 01:21 AM
Opalg

Quote:

Originally Posted by Mauritzvdworm

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 $M_2(\mathbb{C})$ .
March 22nd 2011, 01:35 AM
Mauritzvdworm

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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