Definition 30.1.4 (*-Homomorphism).label Let $A, B$ be $C^{*}$-algebras and $\phi: A \to B$, then $\phi$ is a *-homomorphism if:

1. (1)

   $\phi$ is a homomorphism of Banach algebras.

2. (2)

   For every $x \in A$, $\phi(x^{*}) = \phi(x)^{*}$.

If in addition, $\phi(1) = 1$, then $\phi$ is a unital *-homomorphism.