2025/04/09 by Carter, Max
#22D15 #43A20 #43A45 #46H05 #FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2504.06893
The question of whether the group ℚp \rtimes ℚp^* 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 \mathbbK, the affine group \mathbbK \rtimes \mathbbK^* is a Hermitian group. The proof is a consequence of results about Hermitian Banach *-algebras from the 1970's. In the case that \mathbbK 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.