The Spine of Local Fell Groups

Given a locally compact group \(G\), the spine of the Fourier-Stieltjes Algebra \(A^*(G)\), introduced by M. Ilie and N. Spronk, is a subalgebra of \(B(G)\) which contains all \(A(H) \circ \eta\) where \(\eta : G \to H\) is a continuous homomorphism. We say a group is spinal if \(A^*(G)\) is all of \(B(G)\). Naturally all compact groups are spinal. A known non-compact example is the Fell group \(G = \mathbb{Q}_p \rtimes \mathbb{O}_p^*\), where \(\mathbb{Q}_p\) and \(\mathbb{O}_p\) are the \(p\)-adic numbers and integers respectively. We show that if we replace \(\mathbb{Q}_p\) with a totally disconnected local field, then this group is also spinal. To date, these local Fell groups are the only known non-compact spinal groups. We also explore the higher dimensional analogue \(G = \mathbb{Q}_p^2 \rtimes \mathbb{O}_p^*\), where we compute the spine explicitly. We show in this case that \(G\) is not spinal, though in some sense, it is not much larger than \(A^*(G)\).