Spectral Properties of $C_0$-Semigroups of Positive Operators on C${}^{\ast }$-Algebras

Let $(T(t){)}_{t\ge 0}$ be a positive $C_0$-semigroup with generator $A$ on a C${}^{\ast }$-algebra or on the predual of a W${}^{\ast }$-algebra. Then the growth bound ${\omega }_0$ equals $s(A)$. If the spectrum of $A$ is not empty, then $s(A)$, the spectral bound of $A$, is a spectral value.