The Spine of a Fourier Algebra

Given a locally compact group \(G\), one can define the Fourier and Fourier-Stieltjes algebras \(A(G)\) and \(B(G)\), which in the abelian case, are isomorphic to \(L^1(\hat{G})\) and \(M(\hat{G})\) respectively. The Fourier algebra \(A(G)\) is typically more tractable than \(B(G)\), and often easier to describe. A notable exception is when \(B(G) = A(G)\), which occurs precisely when \(G\) is compact. The spine of a Fourier 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. It has been shown that for \(G = \mathbb{Q}_p \rtimes \mathbb{O}_p^*\), that \(B(G) = A^*(G)\), despite not being compact. We also explore \(G = \mathbb{Q}_p^2 \rtimes \mathbb{O}_p^*\), where we have shown that although \(B(G)\) is strictly larger than \(A^*(G)\), they are close to being similar.