TITLE: Wiener Tauberian Theorem for nonunimodular group
QUESTION [5 upvotes]: Is there a nonunimodular group for which Wiener's Tauberian theorem is true? 
Is a locally compact topological  group whose volume grows polynomially with radius always unimodular?

REPLY [7 votes]: To the first is yes also. The example is already given above: ax+b group. It is the semidirect product $\mathbb R^\times \ltimes \mathbb R$. Leptin has proved in Leptin, H., Ideal theory in group algebras of locally compact groups. Invent.Math. 31 (1975/76), no. 3, 259-278 that every semidirect product of separable abelian l.c. groups has the Wiener property (in the sense that every proper closed two-sided ideal is annihilated by a nonzero *-representation).