A tauberian group algebra

Let G be the group of real matrices

$$(x,y) = \left( {\begin{array}{*{20}{c}} {{e^x}} & 0 \\ y & 1 \\ \end{array} } \right)\quad (x,y \in R).$$

Every proper closed two-sided ideal of ${L^1}(G)$ is contained in a maximal modular two-sided ideal. The strong radical of ${L^1}(G)$ is the set of all $f \in {L^1}(G)$ with $\smallint f(x,y)\;dy = 0$ for almost all $x \in R$. The strong structure spaces of ${L^1}(G)$ and ${L^1}(R)$ are homeomorphic.