Browsing by Author "Westlund, Tim"

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    The Banach-Tarski paradox
    (2022-07-04) Elofsson, Carl; Nilsson, Adrian; Söderberg, Nicolas; Westlund, Tim; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaper
    In this thesis we present a proof of the Banach-Tarski paradox, a counterintuitive result that states that any ball in R3 can be cut into finitely many pieces and then be reassembled into two copies of the original ball. Since the result follows from the axiom of choice it is important for assessing its role as an axiom of mathematics. A related result that we also include is that the minimal number of pieces in such a decomposition of any ball in R3 is five. The proof uses the paradoxicality of the free group on two generators and the existence of a free subgroup of the special orthogonal group SO3. We also give a proof of Tarski’s theorem, which states that the existence of a finitely additive, isometry invariant measure normalizing a set is equivalent to that set not being paradoxical. The proof makes use of the Hahn-Banach theorem and relies on the concept of a group acting on several copies of a set.
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    Similarity Problems: Which Groups Are Unitarizable?
    (2024-08-20) Westlund, Tim; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaper
    This thesis covers some theory on similarity of group representations to unitary representations. We discuss the notion of amenability and give some classes of groups that are amenable. We then prove the Dixmier-Day theorem, that states that a locally compact group G is unitarizable if it is amenable. We also investigate the converse of this statement, which is still an open problem. We will give some statements where we make some assumptions on the similarity that are equivalent to amenability. We will also investigate when bounded algebra homomorphism A → B(H), where A is a C∗-algebra, are similar to a *-homomorphism. We will present connections between the unitarizability of groups and unitarizability of group C∗-algebras, and this will be useful for some results about the converse of the Dixmier-Day theorem. We will also investigate the notions of completely positive and completely bounded maps and prove Stinespring’s theorem for completely positive maps followed by Wittstock’s theorem for completely bounded maps. We then prove Haagerup’s theorem that states that unitarizability of homomorphisms is equivalent to the property of being completely bounded.