On closed Lie ideals and center of generalized group algebras

For any locally compact group $G$ and any Banach algebra $A$, a characterization of the closed Lie ideals of the generalized group algebra $L^1(G,A)$ is obtained in terms of left and right actions by $G$ and $A$. In addition, when $A$ is unital and $G$ is an $[\mathbf{S}\mathbf{I}\mathbf{N}]$ group, we show that the center of $L^1(G,A)$ is precisely the collection of all center valued functions which are constant on the conjugacy classes of $G$. As an application, we establish that $\mathcal{Z}(L^1(G){\otimes }^{\gamma }A)=\mathcal{Z}(L^1(G)){\otimes }^{\gamma }\mathcal{Z}(A)$, for a class of groups and Banach algebras. And, prior to these, for any finite group $G$, the Lie ideals of the group algebra $\mathbb{C}[G]$ are identified in terms of some canonical spaces determined by the irreducible characters of $G$.