Fixed Point Algebras for Easy Quantum Groups

Abstract

Abstract. Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their K-groups. Building on prior work by the second author, we prove that free easy quantum groups satisfy these conditions and we compute the K-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group S + n , the free orthogonal quantum group O + n and the quantum reflection groups H s+ n . Our fixed point-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups, which are related to Hopf-Galois extensions.

Equivariant -correspondences and compact quantum group actions on Pimsner algebras

“…For example, compact quantum groups have been found to preserve fewer KMS states on certain graph C * -algebras as opposed to compact group actions [JM21a]. As a necessarily incomplete list of references for the reader interested in this direction, we mention [Gab14,GW16,Kat17,Pao97].…”

Part III, Free Actions of Compact Quantum Groups on C^*-Algebras

“…Since [11,Proposition 6.10] implies that K 0 (B) can be identified with the integers in such a way that [1 B ] = 1 (see also [12,14,16]), it follows from Lemma 4.10 that (O 2 , SU q (2), α) is not cleft.…”