Injective C-star homomorphism is isometric
= Injective C-star homomorphism is isometric
An injective <C-star homomorphism> preserves norms. For unital maps, reduce to the commutative algebra generated by $1$ and $a^*a$. Its induced map on <character spaces> has dense, hence full, image; otherwise a nonzero continuous function vanishing on that image would lie in the kernel. The <C-star identity> then gives norm preservation for $a$.