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dc.contributor.authorBrannan, Michael Paul
dc.contributor.otherQueen's University (Kingston, Ont.). Theses (Queen's University (Kingston, Ont.))en
dc.date2012-07-27 11:50:51.616en
dc.date2012-07-31 12:45:57.767en
dc.date.accessioned2012-08-03T16:03:08Z
dc.date.available2012-08-03T16:03:08Z
dc.date.issued2012-08-03
dc.identifier.urihttp://hdl.handle.net/1974/7342
dc.descriptionThesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2012-07-31 12:45:57.767en
dc.description.abstractIn this thesis, we study the operator algebraic structure of various classes of unimodular free quantum groups, including thefree orthogonal quantum groups $O_n^+$, free unitary quantum groups $U_n^+$, and trace-preserving quantum automorphism groups associated to finite dimensional C$^\ast$-algebras. The first objective of this thesis to establish certain approximation properties for the reduced operator algebras associated to the quantum groups $\G = O_n^+$ and $U_n^+$, ($n \ge 2$). Here we prove that the reduced von Neumann algebras $L^\infty(\G)$ have the Haagerup approximation property, the reduced C$^\ast$-algebras $C_r(\G)$ have Grothendieck's metric approximation property, and that the quantum convolution algebras $L^1(\G)$ admit multiplier-bounded approximate identities. We then go on to study trace-preserving quantum automorphism groups $\G$ of finite dimensional C$^\ast$-algebras $(B, \psi)$, where $\psi$ is the canonical trace on $B$ induced by the regular representation of $B$. Here, we extend several known results for free orthogonal and free unitary quantum groups to the setting of quantum automorphism groups. We prove that the discrete dual quantum groups $\hG$ have the property of rapid decay, the von Neumann algebras $L^\infty(\G)$ have the Haagerup approximation property, and that $L^\infty(\G)$ is (in most cases) a full type II$_1$-factor. As applications of these and other results, we deduce the metric approximation property, exactness, simplicity and uniqueness of trace for the reduced C$^\ast$-algebras $C_r(\G)$, and the existence of multiplier-bounded approximate identities for the convolution algebras $L^1(\G)$. We also show that when $B$ is a full matrix algebra, $L^\infty(\G)$ is an index $2$ subfactor of $L^\infty(O_n^+)$, and thus solid and prime. Finally, we investigate strong Haagerup inequalities in the context of quantum symmetries arising from actions of free quantum groups on non-commutative random variables. We prove a generalization of the strong Haagerup inequality for $\ast$-free R-diagonal families due to Kemp and Speicher, and apply this result to study strong Haagerup inequalites for the free unitary quantum groups.en_US
dc.languageenen
dc.language.isoenen_US
dc.relation.ispartofseriesCanadian thesesen
dc.rightsThis publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner.en
dc.subjectquantum groupsen_US
dc.subjectoperator algebrasen_US
dc.subjectfree probabilityen_US
dc.subjectapproximation propertiesen_US
dc.titleOn the Reduced Operator Algebras of Free Quantum Groupsen_US
dc.typeThesisen_US
dc.description.degreePh.Den
dc.contributor.supervisorMingo, James A.en
dc.contributor.supervisorSpeicher, Rolanden
dc.contributor.departmentMathematics and Statisticsen


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