Bulletin of the
Korean Mathematical Society BKMS

Using the Hyers--Ulam--Rassias stability method, we investigate isomorphisms in quasi-Banach algebras and derivations on quasi-Banach algebras associated with the Cauchy--Jensen functional equation $$2f(\frac{x+y}{2}+z) = f(x)+f(y)+2f(z),$$ which was introduced and investigated in [2, 17]. The concept of Hyers--Ulam--Rassias stability originated from the Th. M. Rassias' stability theorem that appeared in the paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. $\bf 72$ (1978), 297--300. Furthermore, isometries and isometric isomorphisms in quasi-Banach algebras are studied.

Keywords: Cauchy--Jensen functional equation, isomorphism, isometry, derivation, quasi-Banach algebra

MSC numbers: Primary 46B03, 47Jxx, 47B48