The affine group of a local field is Hermitian

Abstract

The question of whether the group \({\mathbb {Q}}_p \rtimes {\mathbb {Q}}_p^*\) is Hermitian has been stated as an open question in multiple sources in the literature, even as recently as a paper by R. Palma published in 2015. In this note, we confirm that this group is Hermitian by proving the following more general theorem: given any local field \({\mathbb {K}}\), the affine group \({\mathbb {K}} \rtimes {\mathbb {K}}^*\) is a Hermitian group. The proof is a consequence of results about Hermitian Banach \(*\)-algebras from the 1970s. In the case that \({\mathbb {K}}\) is a non-archimedean local field, this result produces examples of totally disconnected locally compact Hermitian groups with exponential growth, and these are the first examples of groups satisfying these properties. This answers a second question of Palma about the existence of such groups.