Aleksa Vujičić, University of Waterloo
PhD Thesis Defence
In 1972, Baggett showed that a locally compact group G is compact if and only if its dual space of irreducible representations 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 Op* ⋊ Qp, where Op denotes the p-adic integers, and Qp the p-adic numbers).
Baggett shows that this is a noncompact whose dual is not countable.
In this talk, we shall discuss the dual space structure of this and other related groups.
It is well known that p-adics are an instance of a local field, that is a non-discrete locally compact field.
In the corresponding thesis, we generalise the results of Baggett to what we call the local Fell groups, the local field equivalents of the Fell group.
We also work in this local context for all forthcoming results, though we state it in terms of p-adics to simplify matters.
This talk is divided into two parts.
In the first, 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) ∘ q_K, where B_∞(G) is the Fourier space generated by purely infinite representations.
We also show that in groups with countable open orbits (such as the local Fell groups) this simplifies further to B(G) = A(G) ⊕ A(K) ∘ q_K.
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.
In the second portion, we study the amenability of the central Fourier algebra ZA(G) = A(G) ∩ L1(G) for G = Op ⋊ Op*.
We show that ZA(G) contains as a quotient the Fourier algebra of a hypergroup, which is induced by the 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 this H satisfies 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.
MC 2009