Nonsingular Bernoulli actions & quantum symmetries

Abstract

This thesis contains the results obtained in the research articles [VW17] [AdLW16] [AdLW17] and [TW16]. We study four types of mathematical phenomena. 1. We prove existence and nonexistence results for type III Bernoulli actions of discrete groups. We show that the existence of a type III Bernoulli action of a given discrete group G depends on the existence of a nonzero element inside its first L 2 -cohomology H1 (G, `2 (G)). We also provide concrete examples of type III Bernoulli actions of Z and of free product groups [VW17]. 2. We define and examine the Fourier algebra, the Fourier-Stieltjes algebra and the algebra of completely bounded multipliers of a rigid C ∗ -tensor category [AdLW17]. 3. We prove that the representation categories of q-deformations of connected compact simple Lie groups with trivial center, 0 < q ≤ 1 have the HoweMoore property [AdLW16]. 4. We describe a correspondence between the free product of planar algebras and the free wreath product of compact quantum groups and examine its consequences [TW16].

@phdthesis{wahlNonsingularBernoulliActionsQuantum2018,
 author = {Jonas Wahl},
 month = {6},
 school = {ARENBERG DOCTORAL SCHOOL Faculty of Science},
 title = {Nonsingular Bernoulli actions & quantum symmetries},
 type = {PhD-Thesis},
 url = {https://www.dfki.de/fileadmin/user_upload/import/17047_thesis_(1).pdf},
 year = {2018}
}