Timeline for The Haar state on compact quantum groups $A_u(Q)$ and $A_o(Q)$

6 events

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Jan 7, 2013 at 12:56	comment	added	Uwe Franz		It might be useful to stress that the Haar state is always faithful on the dense Hopf -algebra spanned by the coefficients of the finite-dimensional corepresentations of a compact quantum groups (which is the -algebra generated by the $u_{ij}$ in these examples) and - by construction - on the reduced $C^*$-algebra.
Jul 12, 2012 at 19:36	vote	accept	m07kl
Mar 16, 2012 at 9:41	comment	added	m07kl		thanks! For universal compact quantum groups, Co-amenability is equivalent to faithfulness of the Haar state (see Thm 3.6 "Co-Amenability of Compact Quantum Groups" by E. B´edos G.J. Murphy L. Tuset). $A_o(n)$ is coamenable only for $n = 2$, while $A_u(n)$ is not coamenable for any $n$.
Mar 16, 2012 at 9:04	comment	added	Tom Cooney		I believe (but I do not at this precise moment have access to the relevant papers, so please check for yourself) that $A_u(1_n)$ and $A_o(1_n)$ are not co-amenable for $n \geq 3$, which would imply that the Haar state is not faithful.
Mar 15, 2012 at 16:41	comment	added	m07kl		Thanks! This paper gives explicit formula for $A_u(1_n)$ and $A_o(1_n)$. They are both tracial, since $\kappa^2=id$. Do you know any thing about faithfulness?
Mar 15, 2012 at 14:41	history	answered	Tom Cooney	CC BY-SA 3.0