On C∗‐completions of discrete quantum group rings

Abstract

We discuss just infiniteness of C * -algebras associated to discrete quantum groups and relate it to the C * -uniqueness of the quantum groups in question, that is, to the uniqueness of a C * -completion of the underlying Hopf * -algebra. It is shown that duals of q-deformations of simply connected semisimple compact Lie groups are never C * -unique. On the other hand, we present an example of a discrete quantum group which is not locally finite and yet is C * -unique.

The Gaussian Part of a Compact Quantum Group

“…those for which the corresponding Hopf algebra is finitedimensional; ‚ Quantum permutation groups, i.e. quantum subgroups of S Ǹ for N P N (this includes for instance the quantum reflection groups H sǸ for all 1 ď s ă `8); ‚ Profinite compact quantum groups in the sense of [CDPR14], or in the terms of [CS19], duals of locally finite discrete quantum groups. Note also that [CDPR14] gives examples of duals of discrete groups which are totally disconnected as compact quantum groups, hence totally strongly disconnected, but not profinite.…”

The RFD and Kac quotients of the Hopf * -algebras of universal orthogonal quantum groups

“…Example 2.8. [7,Proposition 2.4] showed that a compact quantum group is coamenable if and only its reduced C * -algebra admits a finite-dimensional representation. Therefore, using the results of Banica from [1], [2] and [3]…”