Generalized derivations on certain Banach algebras

Let $\mathcal{A}$ be a Banach algebra with the properties that $\mathrm{r}\mathrm{a}\mathrm{d}(\mathcal{A})=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{n}(\mathcal{A})$ and the algebra $\mathcal{A}\mathrm{/}\mathrm{r}\mathrm{a}\mathrm{d}(\mathcal{A})$ is commutative. We show that a derivation of $\mathcal{A}$ maps $\mathcal{A}$ into ${\rm rad}({\mathcal A})$. Using this, we determine among other things when a generalized derivation of $\mathcal{A}$ maps $\mathcal{A}$ into $\mathrm{r}\mathrm{a}\mathrm{d}(\mathcal{A})$. We also study $k$-centralizing generalized derivations of $\mathcal{A}$. Then, for a generalized derivation $(\delta ,d)$ of $\mathcal{A}$ we obtain a necessary and sufficient condition for $({\delta }^2,d^2)$ to be still a generalized derivation of $\mathcal{A}$. The main applications are concerned with the algebras over locally compact groups. In particular, we deduce these results for bidual of Fourier algebras of discrete amenable groups as an application of our approach.