A completion theorem for pro-G-spectra

We give a very general completion theorem for pro-spectra. We show that, if G is a compact Lie group, M [ ∗ ] is a pro-G-spectrum, and F is a family of (closed) subgroups of G, then the mapping pro-spectrum F ( E F + , M [ ∗ ] ) is the F -adic completion of M [ ∗ ] , in the sense that the map M [ ∗ ] → F ( E F + , M [ ∗ ] ) is the universal map into an algebraically F -adically complete pro-spectrum. Here, F ( E F + , M [ ∗ ] ) denotes the pro-G-spectrum { F ( E F + t , M [ s ] ) } s , t , where E F + t runs over the finite subcomplexes of E F + .