dc.contributor.author Brannan, Michael Paul dc.contributor.other Queen's University (Kingston, Ont.). Theses (Queen's University (Kingston, Ont.)) en dc.date 2012-07-27 11:50:51.616 en dc.date 2012-07-31 12:45:57.767 en dc.date.accessioned 2012-08-03T16:03:08Z dc.date.available 2012-08-03T16:03:08Z dc.date.issued 2012-08-03 dc.identifier.uri http://hdl.handle.net/1974/7342 dc.description Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2012-07-31 12:45:57.767 en dc.description.abstract In 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. en_US 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. dc.language en en dc.language.iso en en_US dc.relation.ispartofseries Canadian theses en dc.rights This 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.subject quantum groups en_US dc.subject operator algebras en_US dc.subject free probability en_US dc.subject approximation properties en_US dc.title On the Reduced Operator Algebras of Free Quantum Groups en_US dc.type Thesis en_US dc.description.degree Ph.D en dc.contributor.supervisor Mingo, James A. en dc.contributor.supervisor Speicher, Roland en dc.contributor.department Mathematics and Statistics en
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