Symmetric bi-derivations on certain Banach algebras

Let $A$ be a Banach algebra with a right identity $u$ such that $uA$ is commutative and semisimple. In this paper, we investigate symmetric bi-derivations of $A$ and detremine their range. We also study symmetric bi-derivations of $A$ with their $k$-centralizing trace. Finally, we prove every symmetric Jordan bi-derivation of $A$ is a symmetric bi-derivation.