Distance Distribution Functions and Probabilistic Metric Spaces on C*-algebras

In this paper, by using the concept of positive elements of $C^*$-algebras instead of the real numbers $\mathbb{R}$, a generalization of distribution functions, with a particular focus on distance distribution functions has been introduced as a novel approach in statistical mathematics. Also, by applying this concept, probabilistic metric space on $C^*$-algebras is defined and some properties such as topology and fixed point theorems have been investigated. Finally, some examples and applications to integral equations are given.