34.5 *-Homomorphisms

Definition 34.5.1 (*-Homomorphism).label Let $A, B$ be involutive algebras over $\complex$ and $\phi: A \to B$, then $\phi$ is a *-homomorphism if:

1. (SH1)

   For each $x, y \in A$ and $\lambda \in \complex$, $\phi(\lambda x + y) = \lambda \phi(x) + \phi(y)$.

2. (SH2)

   For each $x, y \in A$, $\phi(xy) = \phi(x)\phi(y)$.

3. (SH3)

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

If $A$ and $B$ are unital, then $\phi$ is unital if:

1. (U)

   $\phi(1_{A}) = \one_{B}$.

Proposition 34.5.2.label Let $A, B$ be unital $C^{*}$-algebras and $\phi: A \to B$ be a unital *-homomorphism, then for each $x \in A$,

1. (1)

   $\sigma_{B}(\phi(x)) \subset \sigma_{A}(x)$.

2. (2)

   $\norm{\phi(x)}_{B} \le \norm{x}_{A}$.