Let $ku$ be the $p$-completion of the connective complex K-theory spectrum. The group $\Bbb{Z}_p^\times\cong \Delta \times \Bbb{Z}_p$ acts on $ku$, where $\Delta$ is the cyclic group of order $p-1$. Let $\ell$ be the connective Adams summand.
On page 4 of this article, it is written that $\ell \simeq ku^{h\Delta}$. Why is that? Where can I find a proof of this result?
Thanks!