C-star homomorphism ($\theta (a^{\ast })=\theta (a{)}^{\ast }$)

A C-star homomorphism is a complex linear multiplicative map between C-star algebras that preserves the involution. Such maps are contractive; an injective one is an isometry.

An injective C-star homomorphism preserves norms. For unital maps, reduce to the commutative algebra generated by $1$ and $a^{\ast }a$. 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 $a$.