A Laplace transform approach to $C$-semigroups on a $\mathcal{T}_{\varepsilon, \lambda}$-complete random normed module

In this paper, we first introduce the notion of the Laplace transform for an abstract-valued function from $[0, \infty)$ to a $\mathcal{T}_{\varepsilon, \lambda}$-complete random normed module $S$. Then, combining respective advantages of the $(\varepsilon, \lambda)$-topology and the locally $L^0$-convex topology on $S$, we prove the differentiability, Post-Widder inversion formula and uniqueness of such a Laplace transform. Second, based on the above work, we establish the Hille-Yosida theorem for an exponentially bounded $C$-semigroup on $S$, considering both the dense and nondense cases of the range of $C$, respectively, which extends and improves several important results. Finally, we also apply such a Laplace transform to abstract Cauchy problems in the random setting.