Contractive unital linear map is positive

Question

Reading Brown Ozawa book on C* algebras they seem to imply any contractive unital linear map between C* algebras is positive. Does this hold true? And if so why? I don't see why something so general would be positive

Answer 1

Yes, this is true. Let us first prove it when the codomain is $\mathbb{C}$: that is, if $A$ is a unital $C^*$-algebra then a contractive unital linear map $\varphi:A\to\mathbb{C}$ is positive (i.e., a state). Let $a\in A$ be positive and observe that $\|a+\lambda\|\leq |\lambda|$ for all $\lambda\in\mathbb{C}$ with $\operatorname{Re}(\lambda)\leq -\|a\|$. Since $|\varphi(a)+\lambda|=|\varphi(a+\lambda)|\leq\|a+\lambda\|$ we also have $|\varphi(a)+\lambda|\leq|\lambda|$ for such $\lambda$. But this implies $\varphi(a)\geq 0$ (clearly we cannot have $\varphi(a)<0$, and if $\varphi(a)$ is not real, choose $\lambda= Ci\varphi(a)$ for real $C$ such that this $\lambda$ satisfies $\operatorname{Re}(\lambda)\leq -\|a\|$ so then $|\varphi(a)+\lambda|=|1+Ci|\cdot|\lambda|$).

Now suppose $A$ and $B$ are unital $C^*$-algebras and $f:A\to B$ is a contractive unital linear map. For any state $\varphi$ on $B$, $\varphi\circ f$ is contractive and unital since $\varphi$ and $f$ are, and hence is also a state by the previous paragraph. Thus if $a\in A$ is positive, $\varphi(f(a))\geq 0$ for all states $\varphi$ on $B$. This implies that $f(a)$ is positive.