The injection and the projection theorem for spectral sets

Abstract

For a closed normal subgroupN of a locally compact groupG view a closed subset$\tilde E$ of Prim_* L ¹ (G/N) as a subsetE of Prim_* L ¹ (G) in the canonical way and writeN ^⊥ for Prim_* L ¹ (G/N) as a subset of Prim_* L ¹ (G); then the injection theorem says: IfE is spectral (i.e. of synthesis), then$\tilde E$ is so; and if$\tilde E$ andN ^⊥ are spectral, thenE is too. In case of a group of polynomial growth with symmetricL ¹-algebra, where smallest idealsj (E) with given hulls exist, it is known thatN ^⊥ is always spectral. For a closed,G-invariant subsetF of Prim_* L ¹ (N) define a closed subsetE of Prim_* L ¹ (G) by$E = \{ \ker \pi ; \ker \pi |{\rm N} \supseteq \mathop \cap \limits_{P \in F} P\}$. Denote by e (I') the ideal generated byC ₀₀ (G)*I', where theG-invariant idealI' ofL ¹ (N) is viewed as a subset of measures onG, then the projection theorem states: IfE is spectral, thenF is so, and ifF is spectral withe (j (F))=j (E) thenE is spectral. All assumptions are fulfilled for instance, ifG andN are of polynomial growth with symmetricL ¹-algebra and eitherSIN-groups or solvable.