# On Completion of \textit{C}*-algebra-valued metric spaces

## TL;DR

This paper explores the structure of $ C $*-algebra-valued metric spaces, showing their relation to cone metric spaces, and establishes their completion theorem along with connections to Hilbert $ C $*-modules.

## Contribution

It introduces the completion theorem for $ C $*-algebra-valued metric and normed spaces, extending the theory and linking it to Hilbert $ C $*-modules.

## Key findings

- $ C $*-algebra-valued metric spaces are cone metric spaces.
- Established the completion theorem for these spaces.
- Connected $ C $*-algebra-valued normed spaces to Hilbert $ C $*-modules.

## Abstract

The concept of a $ C $*-algebra-valued metric space was introduced in 2014. It is a generalization of a metric space by replacing the set of real numbers by a $ C $*-algebra. In this paper, we show that $ C $*-algebra-valued metric spaces are cone metric spaces in some point of view which is useful to extend results of the cone case to $ C $*-algebra-valued metric spaces. Then the completion theorem of $ C $*-algebra-valued metric spaces is obtained. Moreover, the completion theorem of $ C $*-algebra-valued normed spaces is verified and the connection with Hilbert $ C $*-modules, generalized inner product spaces, is also provided.