A construction of some objects in many base cases of an Ausoni-Rognes conjecture

Let $p$ be a prime, $n\ge 1$, $K(n)$ the $n$th Morava $K$-theory spectrum, ${\mathbb{G}}_n$ the extended Morava stabilizer group, and $K(A)$ the algebraic $K$-theory spectrum of a commutative $S$-algebra $A$. For a type $n+1$ complex $V_n$, Ausoni and Rognes conjectured that (a) the unit map $i_n: L_{K(n)}(S^0) \to E_n$fromthe$K(n)$-local sphere to the Lubin-Tate spectrum induces a map $$K(L_{K(n)}(S^0))\wedge v_{n+1}^{-1}{V}_n\to (K(E_n){)}^{h{\mathbb{G}}_n}\wedge v_{n+1}^{-1}{V}_n$$ that is a weak equivalence, where (b) since ${\mathbb{G}}_n$ is profinite, $(K(E_n){)}^{h{\mathbb{G}}_n}$ denotes a continuous homotopy fixed point spectrum, and © ${\pi }_{\ast }(-)$ of the target of the above map is the abutment of a homotopy fixed point spectral sequence. For $n=1$, $p\ge 5$, and $V_1 = V(1)$,wegiveawaytorealizetheabovemapand(c),byprovingthat$i_1$ induces a map $$K(L_{K(1)}(S^0))\wedge v_2^{-1}{V}_1\to (K(E_1)\wedge v_2^{-1}{V}_1{)}^{h{\mathbb{G}}_1},$$ where the target of this map is a continuous homotopy fixed point spectrum, with an associated homotopy fixed point spectral sequence. Also, we prove that there is an equivalence $$(K(E_1)\wedge v_2^{-1}{V}_1{)}^{h{\mathbb{G}}_1}\simeq (K(E_1){)}^{\tilde{h}{\mathbb{G}}_1}\wedge v_2^{-1}{V}_1,$$ where $(K(E_1){)}^{\tilde{h}{\mathbb{G}}_1}$ is the homotopy fixed points with ${\mathbb{G}}_1$ regarded as a discrete group.