Unital maps taking values in abelian C*-algebras

Question

It is known that a bounded linear functional $f$ on a unital C*-algebra $A$ is positive if and only if $f(I)\geqslant 0$. Is the same true for bounded linear operators $T\colon A\to C(X)$ with $T(I) = 1_X$?

Answer 1

It is not true that a linear functional $f\in A^*$ is positive if it is bounded and $f(I)\geq0$. For instance, let $A=M_2(\mathbb C)$ and define $$ f\left(\begin{bmatrix}x&y\\ z&w\end{bmatrix}\right)=x-w/2. $$ The $f(I)=1/2\geq0$, but $f(E_{22})=-1/2$.

What is true is that a linear functional is positive if and only if $f(I)=\|f\|$.

To extend this to bounded linear maps $T:A\to C(X)$, you need to require $T$ to be contractive. The result is that if $T$ is unital and contractive, then it is positive. This works with the codomain any C$^*$-algebra.

Assume $T:A\to B$ is unital and contractive.

For each state $\phi\in B^*$ (i.e. $\phi$ is positive and unital), the functional $\phi_T\in A^*$ given by $\phi_T(a)=\phi(Ta)$ is then unital and contractive, so positive. So, if $a\geq0$, then $\phi(Ta)\geq0$ for all states $\phi$. This implies that $Ta\geq0$. So $T\geq0$.