An Introduction to C∗-Algebra-Valued Metric-Like Space With Related Applications

It is well-known that the main objective of metric-like spaces is to pave way for the development of fixed point results in nonstandard framework, such as in nonsymmetric distance structures, where self-distances might be nonzero. This objective becomes more achievable when the range set of the distance function is a C∗-algebra. Unfortunately, the idea of the latter is yet to be examined, following the existing literature. Hence, the principal focus of this paper is to introduce the concept of C∗-algebra-valued metric-like space. Thereafter, some topological properties and fixed point theorems in the new space are studied. Comparative examples are provided to show that the proposed ideas herein are new and generalize a few important corresponding results in the literature. From application point of view, new conditions for the existence and uniqueness of a solution to a class of integral equations are established, using one of our obtained theorems.