Journal of the
Korean Mathematical Society JKMS

We investigate the continuity of $(\alpha,\beta)$-derivations on $B(\mathfrak{X})$ or $C^*$-algebras. We give some sufficient conditions on which $(\alpha,\beta)$-derivations on $B(\mathfrak{X})$ are continuous and show that each $(\alpha,\beta)$-derivation from a unital $C^*$-algebra into its a Banach module is continuous when $\alpha$ and $\beta$ are continuous at zero. As an application, we also study the ultraweak continuity of $(\alpha,\beta)$-derivations on von Neumann algebras.

Keywords: automatic continuity, ($\alpha, \beta$)-derivation, Banach algebra, $C^*$-algebra

MSC numbers: 46H40, 46L57