Journal of the
Korean Mathematical Society JKMS

We prove the Hyers--Ulam stability of linear $d$-isometries in linear $d$-normed Banach modules over a unital $C^*$-algebra and of linear isometries in Banach modules over a unital $C^*$-algebra. The main purpose of this paper is to investigate $d$-isometric $C^*$-algebra isomorphisms between linear $d$-normed $C^*$-algebras and isometric $C^*$-algebra isomorphisms between $C^*$-algebras, and $d$-isometric Poisson $C^*$-algebra isomorphisms between linear $d$-normed Poisson $C^*$-algebras and isometric Poisson $C^*$-algebra isomorphisms between Poisson $C^*$-algebras.

Keywords: Hyers--Ulam stability, linear $d$-normed Banach module over $C^*$-algebra, isometric isomorphism, $d$-isometric isomorphism, Cauchy additive mapping

MSC numbers: Primary 17B63, 39B52, 46B04, 46L05, 47C15, 51Kxx