Harmonic analysis on totally disconnected locally compact groups
(2026)
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- (en) The harmonic analysis of locally compact groups has been studied for over 100 years, and has been investigated deeply, particularly in the settings of abelian groups, compact groups, and connected Lie groups. While many harmonic analysis properties are well understood for these classes of groups, in contrast, our understanding of them remains significantly underdeveloped in the setting of totally disconnected locally compact (tdlc) groups. This thesis studies harmonic analysis properties coming from unitary representation theory (e.g. the type I and CCR properties), and from Banach algebra theory (e.g. the Wiener, Hermitian and *-regular properties), for many classes of tdlc groups. The following questions underpin the work carried out in this thesis: (i) Do the established results regarding these properties for connected Lie groups have direct analogues for tdlc groups? (ii) How does the dichotomy between amenable and non-amenable groups influence these properties in the tdlc setting? The thesis starts in Chapter 1 with an introduction, where the above listed properties are linked to a central topic, spectral synthesis, which is rooted in classical questions from Fourier analysis. Chapters 2, 3 and 5 investigate Banach algebras on tdlc groups, and in particular, study the Hermitian, Wiener and *-regular properties. It is known that the Banach *-algebra L¹(G) cannot be Hermitian or *-regular for non-amenable groups, and in Chapter 2, we further show that for many non-amenable tdlc groups, L¹(G) is also not Wiener. In Chapter 3, we study locally elliptic groups, which are groups that are ascending unions of compact open subgroups, and hence amenable. For such groups it is known that L¹(G) is Wiener and *-regular whenever it is Hermitian. In Chapter 3, we show that this holds for a much broader class of Banach *-algebras on locally elliptic groups which are *-subalgebras of L¹(G), and develop numerous criterions for determining when these algebras are Hermitian. In Chapter 5, we show that L¹(G) is Hermitian when G is the affine group of a (non-archimedean) local field, resolving a long standing open question. In Chapter 4, we investigate the unitary representation theory of two-step nilpotent locally compact groups, with a focus on tdlc groups and the type I and CCR properties. While Chapters 2, 3 and 5 on Banach algebras show that the analogous results for Lie groups hold almost identically for tdlc groups, Chapter 4 provides some contrasting phenomena. On one hand, we tackle the question of whether unipotent groups over positive characteristic local fields are type I and CCR, and provide positive results in the two-step nilpotent case. On the other hand, we produce uncountably many examples of non-type I p-torsion two-step nilpotent tdlc groups, providing contrasting phenomena to the setting of connected Lie groups where all nilpotent groups are CCR (and hence type I).
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Carter, M. (2026). Harmonic analysis on totally disconnected locally compact groups. https://hdl.handle.net/2078.5/280703