# On Quantum Duality of Group Amenability

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^{†}

## Abstract

**:**

## 1. Introduction

**Definition**

**1**

**.**Assume that A is a C*-algebra with an identity and $\Delta :A\to A\otimes A$ is a unital *-homomorphism satisfying the following two relationships,

- (i)
- $(id\otimes \Delta )\Delta =(\Delta \otimes id)\Delta ,$
- (ii)
- the linear spans of$(1\otimes A)\Delta (A)$and$(A\otimes 1)\Delta (A)$are each equal to$A\otimes A$.

## 2. Preliminaries

**Definition**

**2**

**.**A $CQG$$(A,\Delta )$ is called co-amenable if the co-unit ${\epsilon}_{r}$ of $({A}_{r},{\Delta}_{r})$ is norm-bounded, where $({A}_{r},{\Delta}_{r})$ is the reduced quantum group of $(A,\Delta )$.

**Proposition**

**1**

**.**Let $(A,\Delta )$ be a $CQG$, and h and ε be its Haar integral and co-unit, respectively. Then, $(A,\Delta )$ is co-amenable if and only h is faithful and ε is norm-bounded.

**Proposition**

**2**

**.**Let $(A,\Delta )$ be a $CQG$, $({A}_{0},{\Delta}_{0})$ be the associated $algCQG$ of $(A,\Delta )$, and ${\parallel \xb7\parallel}_{c}$ a regular C*-norm on ${A}_{0}$. Then,

- (i)
- For all$a\in {A}_{0}$,$${\parallel a\parallel}_{r}\le {\parallel a\parallel}_{c}\le {\parallel a\parallel}_{u}.$$
- (ii)
- $(A,\Delta )$is co-amenable if and only if$$(A,\Delta )=({A}_{u},{\Delta}_{u})=({A}_{r},{\Delta}_{r}).$$

**Definition**

**3.**

- (1)
- Let${A}_{0}$and${B}_{0}$be two$algCQGs$, and$<\xb7,\xb7>:{A}_{0}\otimes {B}_{0}\u27f6\mathbb{C}$be a bilinear form. Assume that they satisfy the relations$$\langle \Delta (a),{b}_{1}\otimes {b}_{2}\rangle =\langle a,{b}_{1}{b}_{2}\rangle ,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\langle {a}_{1}\otimes {a}_{2},\Delta (b)\rangle =\langle {a}_{1}{a}_{2},b\rangle ,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\langle {a}^{\ast},b\rangle =\overline{\langle a,{S}_{{B}_{0}}{(b)}^{\ast}\rangle},$$$$\langle a,{1}_{{B}_{0}}\rangle ={\epsilon}_{{A}_{0}}(a),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\langle {1}_{{A}_{0}},b\rangle ={\epsilon}_{{B}_{0}}(b),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\langle {S}_{{A}_{0}}(a),b\rangle =\langle a,{S}_{{B}_{0}}(b)\rangle ,$$
- (2)
- Let$\langle \xb7,\xb7\rangle :A\otimes B\u27f6\mathbb{C}$is a bilinear form. If$({A}_{0},{B}_{0},\langle \xb7,\xb7\rangle {|}_{{A}_{0}\otimes {B}_{0}})$is an algebraic compact quantum group pairing, then the bilinear form is called a compact quantum group pairing, denoted by$(A,B,\langle \xb7,\xb7\rangle )$.

**Proposition**

**3.**

**Proposition**

**4.**

**Definition**

**4.**

## 3. The Main Results

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CQG | Compact Quantum Group |

CQGs | Compact Quantum Groups |

algCQG | algebraic Compact Quantum Group |

DQG | Discrete Quantum Group |

## References

- Woronowicz, S.L. Compact Quantum Groups. In Symétries Quantiques; North-Holland Publishing House: Amsterdam, The Netherlands, 1998; pp. 845–884. [Google Scholar]
- Guo, M.Z.; Jiang, L.N.; Zhao, Y.W. A Pairing Theorem between a Braided Bialgebra and Its Dual Bialgebra. J. Algebr.
**2001**, 245, 532–551. [Google Scholar] - Bédos, E.; Murphy, G.J.; Tuset, L. Co-amenability of compact quantum groups. J. Geom. Phys.
**2001**, 40, 130–153. [Google Scholar] - Bédos, E.; Murphy, G.J.; Tuset, L. Amenaiblity and Co-amenability of algebraic quantum groups. Int. J. Math. Math. Sci.
**2002**, 31, 577–601. [Google Scholar] - Bédos, E.; Murphy, G.J.; Tuset, L. Amenaiblity and Co-amenability of algebraic quantum groups II. J. Funct. Anal.
**2003**, 201, 303–340. [Google Scholar] - Liu, M.; Zhang, X. Quantum double constructions for compact quantum groups. Acta Math. Sin.
**2013**, 29, 2973–2982. [Google Scholar] [CrossRef] - Chen, X. Bekka-type amenability for unitary corepresentations of Locally compact quantum groups. Ann. Funct. Ansl.
**2018**, 2, 210–219. [Google Scholar] [CrossRef] - Abe, E. Hopf Algebras; Cambridge University Press: Cambridge, UK, 1977. [Google Scholar]
- Jiang, L.N.; Guo, M.Z.; Qian, M. The duality theory of a finite dimensional discrete quantum group. Proc. Am. Math. Soc.
**2004**, 132, 3537–3547. [Google Scholar] [CrossRef] - Jiang, L.N.; Li, Z.Y. Representation and duality of finite Hopf C*-algebra. Acta Math. Sin.
**2004**, 47, 1155–1160. (In Chinese) [Google Scholar] - Liu, M.; Zhang, X. Characterization of compact quantum group. Front. Math. China
**2014**, 9, 321–328. [Google Scholar] [CrossRef] - Woronowicz, S.L. Compact matrix pseudogroups. Comm. Math. Phys.
**1987**, 111, 613–665. [Google Scholar] [CrossRef] - Zhang, X.X.; Guo, M.Z. The regular representation and regular covariant representation of crossed products of Woronowicz C*-algebras. Sci. China Ser. A
**2005**, 48, 1245–1259. [Google Scholar] [CrossRef] - Arano, Y. Comparison of unitary duals of Drinfeld doubles and complex semisimple Lie groups. Commun. Math. Phys.
**2017**, 351, 1137–1147. [Google Scholar] [CrossRef] [Green Version] - Guo, M.Z.; Jiang, L.N.; Qian, M. A pairing theorem between multi-parameter bialgebra and its dual bialgebra. Sci. China Ser A
**2001**, 44, 867–876. [Google Scholar] [CrossRef] - Jiang, L.N.; Wang, L.G. C*-structure of quantum double for finite Hopf C*-algebra. J. Beijing Inst. Technol.
**2005**, 14, 328–331. [Google Scholar] - Xin, Q.L.; Jiang, L.N.; Ma, Z.H. The basic construction from the conditional expectation on the quantum double of a finite group. Czechoslovak Math. J.
**2015**, 65, 347–359. [Google Scholar] [CrossRef] [Green Version] - Drinfeld, V.G. Quantum Groups; International Congress of Mathematicians: Berkeley, CA, USA, 1986. [Google Scholar]
- Timmermann, T. An Invitation to Quantum Groups and Duality-From Hopf Aalgebras to Multiplicative Unitaries and Beyond; European Mathematical Society Publishing House: Zürich, Switzerland, 2008. [Google Scholar]

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**MDPI and ACS Style**

Zhang, X.; Liu, M.
On Quantum Duality of Group Amenability. *Symmetry* **2020**, *12*, 85.
https://doi.org/10.3390/sym12010085

**AMA Style**

Zhang X, Liu M.
On Quantum Duality of Group Amenability. *Symmetry*. 2020; 12(1):85.
https://doi.org/10.3390/sym12010085

**Chicago/Turabian Style**

Zhang, Xia, and Ming Liu.
2020. "On Quantum Duality of Group Amenability" *Symmetry* 12, no. 1: 85.
https://doi.org/10.3390/sym12010085