The topology on the primitive ideal space of transformation group 𝐶*-algebras and C.C.R. transformation group 𝐶*-algebras

Abstract

If ( G , Ω ) (G,\Omega ) is a second countable transformation group and the stability groups are amenable then C ∗ ( G , Ω ) {C^ \ast }(G,\Omega ) is C.C.R. if and only if the orbits are closed and the stability groups are C.C.R. In addition, partial results relating closed orbits to C.C.R. algebras are obtained in the nonseparable case. In several cases, the topology of the primitive ideal space is calculated explicitly. In particular, if the stability groups are all contained in a fixed abelian subgroup H H , then the topology is computed in terms of H H and the orbit structure, provided C ∗ ( G , Ω ) {C^ \ast }(G,\Omega ) and C ∗ ( H , Ω ) {C^ \ast }(H,\Omega ) are E H EH -regular. These conditions are automatically met if G G is abelian and ( G , Ω ) (G,\Omega ) is second countable.