The projection theorem for spectral sets (Q1066395)

Let G be a locally compact group with polynomial growth and symmetric \(L^ 1\)-algebra and N a closed normal subgroup of G. Let F be a closed G-invariant subset of \(\Pr im_* L^ 1(N)\) and \(E=\{\ker \pi\); \(\pi\in \hat G\) with \(\pi\) \(| N(k(F))=0\}\). We prove that E is a spectral subset of \(\Pr im_* L^ 1(G)\) if F is spectral. Moreover, we give the following application to the ideal theory of \(L^ 1(G)\). Suppose that, in addition, N is CCR and G/N is compact. Then all primary ideals in \(L^ 1(G)\) are maximal provided all G-orbits in \(\Pr im_* L^ 1(N)\) are spectral.