Unstable Adams operations on $p$–local compact groups

Abstract

A $p$–local compact group is an algebraic object modelled on the $p$–local homotopy theory of classifying spaces of compact Lie groups and $p$–compact groups. In the study of these objects unstable Adams operations are of fundamental importance. In this paper we define unstable Adams operations within the theory of $p$–local compact groups and show that such operations exist under rather mild conditions. More precisely, we prove that for a given $p$–local compact group $\mathcal{G}$ and a sufficiently large positive integer $m$, there exists an injective group homomorphism from the group of $p$–adic units which are congruent to 1 modulo $p^m$ to the group of unstable Adams operations on $\mathcal{G}$.