Compact orbit spaces in Hilbert spaces and limits of edge-colouring models

Let G be a group of orthogonal transformations of a real Hilbert space H. Let R and W be bounded G-stable subsets of H. Let ‖.‖R‖.‖R be the seminorm on H defined by ‖x‖R:=supr∈R|〈r,x〉|‖x‖R:=supr∈R|〈r,x〉| for x∈Hx∈H. We show that if W is weakly compact and the orbit space Rk/GRk/G is compact for each k∈Nk∈N, then the orbit space W/GW/G is compact when WW is equipped with the norm topology induced by ‖.‖R‖.‖R.
As a consequence we derive the existence of limits of edge-colouring models which answers a question posed by Lovász. It forms the edge-colouring counterpart of the graph limits of Lovász and Szegedy, which can be seen as limits of vertex-colouring models.
In the terminology of de la Harpe and Jones, vertex- and edge-colouring models are called ‘spin models’ and ‘vertex models’ respectively.