# Fixed Points of Compact Quantum Groups Actions on Cuntz Algebras

@article{Gabriel2012FixedPO,
title={Fixed Points of Compact Quantum Groups Actions on Cuntz Algebras},
author={Olivier Gabriel},
journal={Annales Henri Poincar{\'e}},
year={2012},
volume={15},
pages={1013-1036}
}
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…

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## References

SHOWING 1-10 OF 61 REFERENCES

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

• Mathematics
• 1992
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

• Mathematics
• 2000
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

• Mathematics
• 2013
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

• Mathematics
• 2009
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

• Mathematics
• 2006
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