An equivalence of results in $C^*$-algebra valued b-metric and b-metric spaces - Nguyen Van Dung; Vo Thi Le Hang; Diana Dolicanin-Djekic (Applied General Topology, 2017)
We construct a $b$-metric from a given $C^*$-algebra-valued $b$-metric and prove some equivalences between them. Then we show that not only fixed point results but also topological properties on $C^*$-algebra-valued $b$-metric spaces may be deduced from certain results in $b$-metric spaces. In particular, every $C^*$-algebra-valued $b$-metric space is metrizable.