Weakly compact homomorphisms from group algebras

A locally compact group $G$ has the weakly compact homomorphism property if every weakly compact homomorphism from the group algebra ${L^1}(G)$ into another Banach algebra $\mathfrak{B}$ has finite-dimensional range. It has been shown that compact and abelian groups have this property. We extend this to a large class of groups including all solvable groups.