Homotopy fixed points for LK(n)(Enlogical andX) using the continuous action

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