Let $X$ denote a complex $C^*$-algebra and $\{Z(t)\}_{t\geq 0}$ is a $C_0$-semigroup of operators on $X$.

Let $x\in X$ satisfy have $x=x^*$ (x is self-adjoint), such that its spectrum satisfies $\sigma(x)\subset [0,\infty)$.

Then under what conditions does it follow that $[Z^*(t)]=[Z(t)]$, and $\sigma(Z(t)x)\subset [0,\infty)$?