  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 ${\bf [SIN]}$ 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 $G$.
