Symmetric bi-derivations and their generalizations on group algebras

Abstract

Here, we investigate symmetric bi-derivations and their generalizations on $L_0^\infty(\frak{G})^*$. For $\kappa\in{\Bbb N}$, we show that if $B: L_0^\infty(\frak{G})^*\times L_0^\infty(\frak{G})^*\rightarrow L_0^\infty(\frak{G})^*$ is a symmetric bi-derivation such that $[B(m, m), m^\kappa]\in\hbox{Z}(L_0^\infty(\frak{G})^*)$ for all $m\in L_0^\infty(\frak{G})^*$, then $B$ is the zero map. In the case where, $B$ is a symmetric generalized bi-derivation, we characterize it. We also prove that any symmetric Jordan bi-derivation on $L_0^\infty(\frak{G})^*$ is a symmetric bi-derivation.