Fixed Points of Compact Quantum Groups Actions on Cuntz Algebras

  title={Fixed Points of Compact Quantum Groups Actions on Cuntz Algebras},
  author={Olivier Gabriel},
  journal={Annales Henri Poincar{\'e}},
Given an action of a Compact Quantum Group (CQG) on a finite dimensional Hilbert space, we can construct an action on the associated Cuntz algebra. We study the fixed point algebra of this action, using Kirchberg classification results. Under certain conditions, we prove that the fixed point algebra is purely infinite and simple. We further identify it as a C*-algebra, compute its K-theory and prove a “stability property”: the fixed points only depend on the CQG via its fusion rules. We apply… 

Fixed Point Algebras for Easy Quantum Groups

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

Actions of compact quantum groups

These lecture notes, prepared for the summer school "Topological quantum groups", Bedlewo 2015, deal with aspects of the theory of actions of compact quantum groups on C*-algebras ('locally compact

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

We study and classify free actions of compact quantum groups on unital C*-algebras in terms of generalized factor systems. Moreover, we use these factor systems to show that all finite coverings of

Coactions of Hopf $C^*$-algebras on Cuntz-Pimsner algebras

Unifying two notions of an action and coaction of a locally compact group on a $C^*$-cor\-re\-spond\-ence we introduce a coaction $(\sigma,\delta)$ of a Hopf $C^*$-algebra $S$ on a



Some remarks on actions of compact matrix quantum groups on C*-algebras

We construct an action of a compact matrix quantum group on a Cuntz algebra or a UHF-algebra, and investigate the fixed point subalgebra of the algebra under the action. Especially we cnsider the

Coactions of Hopf algebras on Cuntz algebras and their fixed point algebras

We study coactions of Hopf algebras coming from compact quantum groups on the Cuntz algebra. These coactions are the natural generalization to the coalgebra setting of the canonical representation of

Quantum Group Actions on the Cuntz Algebra

Abstract.The Cuntz Algebra carries in a natural way the structure of a module algebra over the quantized universal enveloping algebra Uq(g), and the structure of a co-module algebra over the quantum

A Construction of Finite Index C*-algebra Inclusions from Free Actions of Compact Quantum Groups

Given an action of a compact quantum group on a unital C u -algebra, one can amplify the action with an adjoint representation of the quantum group on a finite dimensional matrix algebra, and

Actions of compact quantum groups on *-algebras

In this paper we investigate a structure of the fixed point algebra under an action of compact matrix quantum group on a C*-algebra B3. We also show that the categories of C-comodules in B and inner

Fusion rules for representations of compact quantum groups

We give a survey of some recent results on the fusion semirings of compact quantum groups (computations of and applications to discrete quantum groups) by using the following simplifying terminology:

Crossed products of Cuntz algebras by quasi-free automorphisms

By the recent classification theorems of Kirchberg and Phillips [15, 22], a certain class of purely infinite simple C∗-algebras is now classified by K-theoretic data. This class includes inductive

Ergodic actions of compact matrix pseudogroups on $C^*$-algebras

Let G be a compact group acting on a unital C∗-algebra M . The action is said to be ergodic if the fixed point algebra MG reduces to scalars. The first breakthrough in the study of such actions was

A Duality Theorem for Ergodic Actions of Compact Quantum Groups on C*-Algebras

The spectral functor of an ergodic action of a compact quantum group G on a unital C*-algebra is quasitensor, in the sense that the tensor product of two spectral subspaces is isometrically contained