Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-682212220211101$C^*$-metric spaces19911996533010.22075/ijnaa.2020.20839.2205ENMahdi MowlaviDepartment of Mathematics,Faculty of Science,Science and Research Branch Islamic Azad University,Tehran , Iran.Madjid MirzavaziriDepartment of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, MashhadMohammadreza MardanbeigiDepartment of Mathematics, Faculty of Science, Science and Research Branch Islamic Azad University, Tehran, IranJournal Article20200710The 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 $(X,\tau_{\delta})$.https://ijnaa.semnan.ac.ir/article_5330_1ba8b26200a89ba63930f1ea47b3263b.pdf