$C^*$-metric spaces

Document Type : Research Paper


1 Department of Mathematics,Faculty of Science,Science and Research Branch Islamic Azad University,Tehran , Iran.

2 Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad

3 Department of Mathematics, Faculty of Science, Science and Research Branch Islamic Azad University, Tehran, Iran


‎The purpose of this article is to introduce the notion of an $\mathfrak{A}$-meter‎, ‎as an operator-valued distance mapping on a set $X$ and investigating the theory of $\mathfrak{A}$-metric spaces‎, ‎where $\mathfrak{A}$ is a noncommutative $C^*$-algebra‎. ‎We demonstrate that each metric space may be seen as an $\mathfrak{A}$-metric space and that every $\mathfrak{A}$-metric space $(X,\delta)$ can be regarded as a topological space $(X,\tau_{\delta})$‎.