I want to prove the next theorem:
If $\pi: A \rightarrow B$ is a star homomorphism, meaning it's an algebra homomorphism which also satisfies: $\pi(x^*)=(\pi(x))^*$, where $A$ is an involutive Banach Algebra, and $B$ a C*-algebra, then $||\pi(x)|| \leq ||x||$
I am given a hint to prove, which I am not sure how to prove it.
The hint says to prove that: $\sigma_{B_{I}} (\pi(y)) \subset \sigma_{A_{I}} (y); \forall y \in A$
where $A_I=A\oplus \mathbb{C}$
Here's what I tried thusfar, I want to prove the converse of this inculsion: Take $\lambda \in (\sigma_{A_I}(y))^c$, thus for $\lambda e -y=b$ is invertible, i.e there exists $b^{-1}$, now $\pi(b^{-1})=(\pi(b))^{-1}=(\pi(b))^* = \pi(b^*)= \bar{\lambda} e_B - \pi(y)^*$, and I want to show that $\lambda e_B -\pi(y)$ is invertible in $B$, but I don't see how $\pi(b)^{-1}=\bar{\lambda} e_B - \pi(y^*)$ is an inverse of $\pi(b)=\lambda e_B -\pi(y)$.
Anyone can enlighten my eyes?
Thanks.