The projection theorem (Q2771244)

Let \(G\) be a locally compact group and \(N\) a closed normal subgroup of \(G\). The author proves a projection theorem for two-sided closed ideals in the convolution algebra \(L^1(G,w)\), where \(w:G\to \mathbb {R}_+\) is a weight on \(G\). The author shows that the mapping \(r\) from the set \({\mathcal I}^{L^\infty(G/N)}\) of the closed two-sided ideals of \(L^1(G,w)\) which are invariant under point-wise multiplication by elements of \(L^\infty(G/N)\) into the set \({\mathcal I^G}\) of two-sided closed \(G\)-invariant ideals in \(L^1(N,w_{|N})\) defined by \(r(I)=\overline{\{f_{|N}; f\in {\mathcal K}(G)*I\}}\) (where \({\mathcal K}(G)\) denotes the space of the continuous compactly supported functions on \(G\)), is an order preserving bijection, whose inverse \(e\) is defined by \(e(J)= \overline{j*{\mathcal K}(G)}.\) This paper generalizes to weighted algebras the results of \textit{W. Hauenschild} and \textit{J. Ludwig} [Monatsh. Math. 92, 167-177 (1981; Zbl 0458.43009)].NEWLINENEWLINEFor the entire collection see [Zbl 0969.00047].