\(p\)-pseudomeasures and closed subgroups
For a real number \(p\) larger than 1 and a locally compact group \(G\), let \(PM_p (G)\) be the set of all \(p\)-pseudomeasures on \(G\). We essentially prove the existence of a conditional expectation from \(PM_p (G)\) onto \(PM_p (H)\), if \(H\) is a closed normal subgroup of \(G\). As a first application we give a natural proof of the statement, due to C. Herz, that \(H\) is a set of \(p\)-synthesis in \(G\). Our approach completely avoids the use of Borel sections. We also prove that for every bounded measure \(\mu\) on \(G\), considered as a \(p\)-pseudomeasure, the following inequality holds: \(||\mu ||_p \geq ||\text{Res}_H \mu ||_p\). Recall that \(PM_p (G)\) is the dual of the Figà-Talamanca Herz algebra \(A_p (G)\). For \(G\) abelian, \(PM_2 (G)\) is isomorphic to \(L^\infty (\widehat G)\) and \(A_2 (G)\) to \(L^1 (\widehat G)\), where \(\widehat G\) is the dual group of \(G\). In analogy with a well-known result of Reiter concerning closed ideals of \(L^1(\widehat G)\), we show as a third application that \(\{\text{Res}_H u |u \in I\}\) is a closed ideal of \(A_p (H)\) for any closed ideal \(I\) of \(A_p (G)\), here \(H\) is supposed to be a closed normal amenable subgroup of \(G\).
scientific article; zbMATH DE number 733129
scientific article