Norm of a unital homomorphism

Question

Let $\mathcal{A}_1$ and $\mathcal{A}_2$ be two unital $C^*$-algebras and $\varphi : \mathcal{A}_1 \to \mathcal{A}_2$ a unital $*$-homomorphism, i.e. a linear map such that $\varphi(xy)=\varphi(x)\varphi(x)$, $\varphi(1_{\mathcal{A}_1})=1_{\mathcal{A}_2}$ and $\varphi(x^*)=\varphi(x)^*$.

Is it true that the norm of $\varphi$ in $\mathcal{B}(\mathcal{A}_1, \mathcal{A}_2)$ is equal to $1$?

I can easily show that $\|\varphi\|\geq 1$ but I am having trouble to see why we should always have that $\|\varphi(x)\|_{\mathcal{A}_2} \leq \|x\|_{\mathcal{A}_1}$ for all $x \in \mathcal{A}_1$.

Answer 1

Any $*$-homomorphism between C$^*$-algebras is contractive. This is standard (i.e., it appears in every book on the subject) and is due to three things:

1. The C $^*$-identity $\|a\|^2=\|a^*a\|$, which reduces the problem to norms of positives;

2. The equality $\|a\|=\text {spr}\, (a) $ for $a $ positive;

3. The fact that a $*$-homomorphism reduces the spectral radius.