Amenability of the Central Fourier Algebra
I gave this talk at Victoria University of Wellington, my undergraduate/Masters university. I cover the non-amenability of the central Fourier algebra of the compact \(p\)-adic motion group \(\mathbb{O}_p \rtimes \mathbb{O}^∗_p\), which was the first research project of my PhD program at Waterloo.
In a result due to Alaghmandan and Spronk, it was shown for a compact group \(G\), that if \(G\) is virtually abelian (that is, has an abelian subgroup of finite index), then a certain algebra ‘\(ZA(G)\)’ is amenable. It is conjectured that these conditions are equivalent. We show a result that the p-adic group \(\mathbb{O}_p\) admits an algebra which is not amenable. This algebra can be identified with a quotient of \(ZA(\mathbb{O}_p \rtimes \mathbb{O}^∗_p)\), which further reinforces the conjecture.