# Math Help - unital *-homomorphism

1. ## 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.

2. 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 $M_2(\mathbb{C})$.

3. Aha, I see. Since $M_2(\mathbb{C})$ is simple it only has two closed two sided ideals, 0 and itself.