Completion of C*-algebra-valued Metric Spaces

Abstract

The concept of a C*-algebra-valued metric space was introduced in 2014. It is a generalization of a metric space 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.