A continuous $p$-adic action on the $K(2)$-local algebraic $K$-theory of $p$-adic complex $K$-theory

Let $p$ be a prime, let $K{U}_p$ be $p$-complete complex $K$-theory, and let ${\mathbb{Z}}_p^{\times }$ denote the group of units in the $p$-adic integers. The $p$-adic Adams operations induce an action of the profinite group ${\mathbb{Z}}_p^{\times }$ on $K{U}_p$, and hence, on the algebraic $K$-theory spectrum $K(K{U}_p)$. For $p\ge 5$, we give an elementary construction of the continuous homotopy fixed point spectrum $(L_{K(2)}K(K{U}_p){)}^{hG}$, where $K(2)$ is the second Morava $K$-theory and $G$ is any closed subgroup of ${\mathbb{Z}}_p^{\times }$. Also, for each $G$, we show that there is an associated strongly convergent homotopy fixed point spectral sequence whose $E_2$-term is given by Jannsen’s continuous group cohomology, with $E_2^{s,\ast }=0$, for all $s>2$. This work is related to a conjecture of Ausoni and Rognes.