Logarithmic topological Hochschild homology of topological $K$-theory spectra

Abstract

In this paper we continue our study of logarithmic topological Hochschild homology. We show that the inclusion of the connective Adams summand $\ell$ into the $p$-local complex connective $K$-theory spectrum $k{u}_{(p)}$, equipped with suitable log structures, is a formally log THH-étale map, and compute the $V(1)$-homotopy of their logarithmic topological Hochschild homology spectra. As an application, we recover Ausoni's computation of the $V(1)$-homotopy of the ordinary THH of $ku$.