Iwasawa theory for $K(1)$-local spectra

The Iwasawa algebra $\Lambda$ is a power series ring in one variable over the $p$-adic integers. It has long been studied by number theorists in the context of $\mathbb{Z}_p$-extensions of number fields. It also arises, however, as a ring of operations in $p$-adic topological $K$-theory. In this paper we study $K(1)$-local stable homotopy theory using the structure theory of modules over the Iwasawa algebra. In particular, for $p$ odd we classify $K(1)$-local spectra up to pseudo-equivalence (the analogue of pseudo-isomorphism for $\lambda$-modules) and give an Iwasawa-theoretic classification of the thick subcategories of the weakly dualizable spectra.