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Title:  On the Reduced Operator Algebras of Free Quantum Groups 
Authors:  Brannan, Michael Paul 

Keywords:  quantum groups operator algebras free probability approximation properties 
Issue Date:  3Aug2012 
Series/Report no.:  Canadian theses 
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 tracepreserving 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 multiplierbounded approximate identities.
We then go on to study tracepreserving 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 multiplierbounded 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 noncommutative random variables. We prove a generalization of the strong Haagerup inequality for $\ast$free Rdiagonal families due to Kemp and Speicher, and apply this result to study strong Haagerup inequalites for the free unitary quantum groups. 
Description:  Thesis (Ph.D, Mathematics & Statistics)  Queen's University, 20120731 12:45:57.767 
URI:  http://hdl.handle.net/1974/7342 
Appears in Collections:  Queen's Graduate Theses and Dissertations Department of Mathematics and Statistics Graduate Theses

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