C*-algebra-valued 𝔪ℛ metric space and fixed point results with application

The primary aim of this paper is to address and overcome certain limitations inherent in Banach’s fixed point theorem, as well as those identified in more recent research. Our objective is to relax several key restrictions associated with fixed point results, including the continuity of the mappings, the completeness of metric spaces, and the contraction conditions. We introduce a generalized notion of the distance function, referred to as the [Formula: see text]-algebra valued [Formula: see text] metric, and establish its fundamental properties. We analyze the topological properties of this new metric space, including convergence of sequences, and completeness. Furthermore, we generalize some well-known contraction mappings and prove fixed point theorems on the [Formula: see text]-complete [Formula: see text]-algebra valued [Formula: see text] metric space (not necessarily complete in metric sense). Our findings extend various fixed point results in the literature to a broader context. Moreover, we provide examples where Banach-type contractions yield the desired results in this new structure, which is not the case in various generalized metric spaces. Finally, we utilize our findings to establish the existence and uniqueness of solutions for an operator equation.