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DTSTART;TZID=America/Toronto:20250930T140000
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URL:https://uwaterloo.ca/pure-mathematics/events/phd-thesis-defence-39
SUMMARY:PhD Thesis Defence
CLASS:PUBLIC
DESCRIPTION:ALEKSA VUJIČIĆ\, UNIVERSITY OF WATERLOO\n\n_PhD Thesis Defenc
 e_\n\nIn 1972\, Baggett showed that a locally compact group G is compact i
 f\nand only if its dual space of irreducible representations G^ is\ndiscre
 te.\n\nCuriously however\, there are non-discrete groups whose duals are\n
 compact\, and such a group was identified in the same paper.\n\nIn a simil
 ar vein\, one can define the Fell group Op* ⋊ Qp\, where Op\ndenotes the
  p-adic integers\, and Qp the p-adic numbers).\n\nBaggett shows that this 
 is a noncompact whose dual is not countable.\n\nIn this talk\, we shall di
 scuss the dual space structure of this and\nother related groups.\n\nIt is
  well known that p-adics are an instance of a local field\, that\nis a non
 -discrete locally compact field.\n\nIn the corresponding thesis\, we gener
 alise the results of Baggett to\nwhat we call the local Fell groups\, the 
 local field equivalents of the\nFell group.\n\nWe also work in this local 
 context for all forthcoming results\, though\nwe state it in terms of p-ad
 ics to simplify matters.\n\nThis talk is divided into two parts.\n\nIn the
  first\, we analyse the Fourier and Fourier-Stieltjes algebras of\nthese l
 ocal Fell groups\, which are of the form A ⋊ K for A abelian\nand K comp
 act.\n\nThese local Fell groups fall into a particular class of groups ind
 uced\nby actions for which the stabilisers are 'minimal'\, and we call suc
 h\ngroups cheap groups.\n\nFor groups of this form\, we show that B(G) = B
 _∞(G) ⊕ A(K) ∘\nq_K\, where B_∞(G) is the Fourier space generated 
 by purely infinite\nrepresentations. \n\nWe also show that in groups with 
 countable open orbits (such as the\nlocal Fell groups) this simplifies fur
 ther to B(G) = A(G) ⊕ A(K) ∘\nq_K.\n\nIn an attempt to generalise this
  to higher dimensional analogues\, for\nwhich the above does not hold true
 \, we examine the structure of\nB_∞(G).\n\nIn particular\, we obtain a r
 esult for dimension two in terms of the\nprojective space\, and we show th
 at this is in some sense the 'best'\ndecomposition that can be made.\n\nIn
  the second portion\, we study the amenability of the central Fourier\nalg
 ebra ZA(G) = A(G) ∩ L1(G) for G = Op ⋊ Op*.\n\nWe show that ZA(G) cont
 ains as a quotient the Fourier algebra of a\nhypergroup\, which is induced
  by the action of Op* ↷ Op.\n\nIn general\, if H is a hypergroup induced
  by an action K ↷ A\, then\nthere is a corresponding dual hypergroup H^ 
 by the dual action.\n\nWhen this is the case\, we show that this H satisfi
 es A(H) = L1(H^)\,\nmimicking the classical result for groups.\n\nWe also 
 show that if H^ has orbits which 'grow sufficiently large'\,\nthen via a r
 esult of Alaghmandan\, the algebra L1(H^) is not amenable.\n\nIn particula
 r\, this shows that ZA(G) is also not amenable\, reaffirming\na conjecture
  of Alaghmandan and Spronk.\n\nMC 2009
DTSTAMP:20260504T111513Z
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