A simple proof of the fixed point theorem in $C^*$-algebra valued metric space

TL;DR

This paper presents a straightforward proof of the fixed point theorem within $C^*$-algebra valued metric spaces, utilizing a fundamental inequality for contractions.

Contribution

It introduces a simple proof method for the fixed point theorem in $C^*$-algebra valued metric spaces based on a new fundamental inequality.

Findings

Established a fundamental inequality for contractions in $C^*$-algebra valued metric spaces.

Provided a simplified proof of the fixed point theorem in this setting.

Demonstrated the applicability of the inequality to fixed point results.

Abstract

We obtain a fundamental inequality for a contraction with respect to a $C^*$-algebra valued metric space. As an application of this inequality a simple proof is given for the fixed point theorem in $C^*$-algebra valued metric space.