Wiener Tauberian Theorem for nonunimodular group - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T09:22:50Z http://mathoverflow.net/feeds/question/84881 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84881/wiener-tauberian-theorem-for-nonunimodular-group Wiener Tauberian Theorem for nonunimodular group spr 2012-01-04T12:59:29Z 2012-01-05T10:27:54Z <p>Is there a nonunimodular group for which Wiener's Tauberian theorem is true? </p> <p>Is a locally compact topological group whose volume grows polynomially with radius always unimodular?</p> http://mathoverflow.net/questions/84881/wiener-tauberian-theorem-for-nonunimodular-group/84919#84919 Answer by Alain Valette for Wiener Tauberian Theorem for nonunimodular group Alain Valette 2012-01-04T23:43:26Z 2012-01-04T23:43:26Z <p>The answer to the second question is yes. Let us show that a non-unimodular, locally compact group $G$ cannot have polynomial growth. Let $\mu$ be left Haar measure, $\Delta$ be the modular function, so that $\mu(Ag)=\mu(A)\Delta(g)$ for $A$ a Borel subset in $G$. Now take for $A$ a compact neighborhood of identity, and $g\in G$ such that $\Delta(g)>1$. Then $Ag^n\subset (Ag)^n$, so $\mu(A)\Delta(g)^n\leq\mu((Ag)^n)$, hence if $\Omega$ is a compact neighborhood of identity containing $Ag$, we have $\mu(A)\Delta(g)^n\leq\mu(\Omega^n)$, and the sequence $(\mu(\Omega^n))_{n\geq 1}$ has exponential growth.</p> http://mathoverflow.net/questions/84881/wiener-tauberian-theorem-for-nonunimodular-group/84943#84943 Answer by Yulia Kuznetsova for Wiener Tauberian Theorem for nonunimodular group Yulia Kuznetsova 2012-01-05T10:27:54Z 2012-01-05T10:27:54Z <p>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).</p>