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Let $p$ be a prime, let $KU_p$ 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 $KU_p$, and hence, on the algebraic $K$-theoryspectrum $K(KU_p)$. For $p \geq 5$, we give an elementary construction of thecontinuous homotopy fixed point spectrum $(L_{K(2)}K(KU_p))^{hG}$, 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 associatedstrongly convergent homotopy fixed point spectral sequence whose $E_2$-term isgiven by Jannsen's continuous group cohomology, with $E_2^{s,\ast} = 0$, forall $s > 2$. This work is related to a conjecture of Ausoni and Rognes.

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