= Unital contraction positivity criterion
{title2=$\|T\|\leq1,\quad T1=1\Longrightarrow T\text{ is positive}$}
A real-linear contraction on $C(X)$ that preserves $1$ is positive: for $0\leq f\leq1$, $\|1-Tf\|_\infty\leq1$ forces $Tf\geq0$, then scale. In the complex case, a unital norm-one functional is real on real functions, since $|1+it\phi(g)|\leq\|1+itg\|_\infty=1+O(t^2)$ for both signs of $t$. The same <positivity> argument applies pointwise to $\phi(f)=Tf(x)$. This turns a unital <contraction semigroup> into a <Feller semigroup>.
Back to article page
