Fourier Analysis of Local Fell Groups

Abstract

In 1972, Bagget showed that a separable locally compact group G is compact if and only if its dual space G^ is discrete. Curiously however, there are non-discrete groups whose duals are compact, and such a group was identified in the same paper. In a similar vein, one can define the “Fell group”, a semidirect product of the units of the p-adic integers 𝕆ₚ* acting via multiplication on the p-adic numbers ℚₚ, which Baggett shows is a noncompact group whose dual is not countable. This Fell group forms a basis of the novel work presented in this thesis.

In Chapters 3 and 4, we look at the more general setting of the p-adic integers and numbers, known respectively as discrete valuation rings (DVRs) and local fields. We compile many known results about these objects, in order to generalise the theory of the Fell group to what we call the “local Fell groups”. While this is primarily background material from a variety of sources, there is additional work required to extend these results so that the theory is coherent and complete. We also briefly study finite-dimensional vector spaces over local fields.

In Chapter 5, we analyse the Fourier and Fourier-Stieltjes algebras of these local Fell groups, which are of the form A ⋊ K for A abelian and K compact. These local Fell groups fall into a particular class of groups induced by actions for which the stabilisers are ‘minimal’, and we call such groups “cheap groups”. For groups of this form, we show that B(G) = B∞(G) ⊕ A(K) ∘ qK, where B∞(G) is the Fourier space generated by purely infinite representations. We also show that in group with countable open orbits (such as the local Fell groups) this simplifies further to B(G) = A(G) ⊕ A(K) ∘ qK. In an attempt to generalise this to higher-dimensional analogues, for which the above does not hold true, we examine the structure of B∞(G). In particular, we obtain a result for dimension two in terms of the projective space, and we show that this is in some sense the ‘best’ decomposition that can be made.

Finally in Chapter 6, we study the amenability of the central Fourier algebra ZA(G) = A(G) ∩ ZL1(G) for G = 𝕆ₚ ⋊ 𝕆ₚ and its local field equivalents. We show that ZA(G) contains as a quotient the Fourier algebra of a hypergroup, which is induced by action of 𝕆ₚ* ↷ 𝕆ₚ. In general, if H is a hypergroup induced by an action K ↷ A, then there is a corresponding dual hypergroup H^ by the dual action. When this is the case, we show that these satisfy A(H) ≅ L1(H^), mimicking the classical result for groups. We also show that if H^ has orbits which ‘grow sufficiently large’, then via a result of Alaghmandan, the algebra L1(H^) is not amenable. In particular, this shows that ZA(G) is also not amenable, reaffirming a conjecture of Alaghmandan and Spronk.