Homotopy fixed points for LK(n)(En∧X)LK(n)(En∧X) using the continuous action

Let K(n)K(n) be the nth Morava K  -theory spectrum. Let EnEn be the Lubin–Tate spectrum, which plays a central role in understanding LK(n)(S0)LK(n)(S0), the K(n)K(n)-local sphere. For any spectrum X  , define E∨(X)E∨(X) to be LK(n)(En∧X)LK(n)(En∧X). Let G   be a closed subgroup of the profinite group GnGn, the group of ring spectrum automorphisms of EnEn in the stable homotopy category. We show that E∨(X)E∨(X) is a continuous G  -spectrum, with homotopy fixed point spectrum (E∨(X))hG(E∨(X))hG. Also, we construct a descent spectral sequence with abutment π*((E∨(X))hG)π*((E∨(X))hG).