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Please use this identifier to cite or link to this item:
http://hdl.handle.net/1974/7342
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| Title: | On the Reduced Operator Algebras of Free Quantum Groups |
| Authors: | Brannan, Michael Paul |
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| Keywords: | quantum groups
operator algebras
free probability
approximation properties |
| Issue Date: | 3-Aug-2012 |
| 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 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. |
| Description: | Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2012-07-31 12:45:57.767 |
| URI: | http://hdl.handle.net/1974/7342 |
| Appears in Collections: | Queen's Theses & Dissertations
Mathematics & Statistics Graduate Theses
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