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 $\sigma(Z(t)x)\subset [0,\infty)$? 



One-parameter semigroup of operators of a $C^*$-algebra applied to positive self-adjoint element
Apart from the fact that I do not understand some parts of your question, self-adjoint elements with positive spectrum define a positive cone in your $C^\ast$ algebra. Positivity-preserving semigroups in $C^\ast$ and von Neumann algebras were extensively studies, you should consult the chapter written by Ulrich Groh in the book


[W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. Lotz, U. Moustakas, R. Nagel (ed.), F. Neubrander, U. Schlotterbeck
One-parameter Semigroups of Positive Operators
Springer, 1986.](http://www.fa.uni-tuebingen.de/research/publications/one-parameter-semigroups-of-positive-semigroups/Nagel%20One-parameter%20Semigroups%20of%20Positive%20Operators.pdf)


