Illinois Journal of Mathematics

A note on reduced and von Neumann algebraic free wreath products

Jonas Wahl

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Abstract

We study operator algebraic properties of the reduced and von Neumann algebraic versions of the free wreath products $\mathbb{G}\wr_{*}S_{N}^{+}$, where $\mathbb{G}$ is a compact matrix quantum group. Based on recent results on their corepresentation theory by Lemeux and Tarrago in [Lemeux and Tarrago (2014)], we prove that $\mathbb{G}\wr_{*}S_{N}^{+}$ is of Kac type whenever $\mathbb{G}$ is, and that the reduced version of $\mathbb{G}\wr_{*}S_{N}^{+}$ is simple with unique trace state whenever $N\geq8$. Moreover, we prove that the reduced von Neumann algebra of $\mathbb{G}\wr_{*}S_{N}^{+}$ does not have property $\Gamma$.

Article information

Source
Illinois J. Math. Volume 59, Number 3 (2015), 801-817.

Dates
Received: 11 February 2016
Revised: 22 March 2016
First available in Project Euclid: 30 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1475266409

Mathematical Reviews number (MathSciNet)
MR3554234

Zentralblatt MATH identifier
1355.46056

Subjects
Primary: 46L54: Free probability and free operator algebras

Citation

Wahl, Jonas. A note on reduced and von Neumann algebraic free wreath products. Illinois J. Math. 59 (2015), no. 3, 801--817. https://projecteuclid.org/euclid.ijm/1475266409


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References

  • S. Baaj and G. Skandalis, Unitaires multiplicatifs et dualité pour es produits croisés de $C^*$-algèbres, Ann. Sci. Éc. Norm. Supér. (4) 26 (1993), 425–488.
  • T. Banica, Le groupe quantique compact libre $U(n)$, Comm. Math. Phys. 190 (1997), no. 1, 143–172.
  • T. Banica and R. Speicher, Liberation of orthogonal Lie groups, Adv. Math. 222 (2009), 1461–1501.
  • J. Bichon, Quantum automorphism groups of finite graphs, Proc. Amer. Math. Soc. 131 (2003), no. 3, 665–673.
  • J. Bichon, Free wreath product by the quantum permutation group, Algebr. Represent. Theory 7 (2004), no. 4, 343–362.
  • M. Brannan, Approximation properties for free orthogonal and free unitary quantum groups, J. Reine Angew. Math. 2012 (2012), 223–251.
  • M. Brannan, Reduced operator algebras of trace-preserving quantum automorphism groups, Doc. Math. 18 (2013), 1349–1402.
  • C. Köstler and R. Speicher, A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with almagamation, Comm. Math. Phys. 291 (2009), 473–490.
  • F. Lemeux, The fusion rules of some free wreath product quantum groups and applications, J. Funct. Anal. 267 (2014), 2507–2550.
  • F. Lemeux and P. Tarrago, Free wreath product quantum groups: the monoidal category, approximation properties and free probability, J. Funct. Anal. 270 (2016), 3828–3883.
  • R. T. Powers, Simplicity of the $C^*$-algebra associated with the free group on two generators, Duke Math. J. 42 (1975), 151–156.
  • T. Timmermann, An invitation to quantum groups and duality: From Hopf algebras to multiplicatives and beyond, EMS Textbooks in Mathematics, EMS, Zürich, 2008.
  • S. Vaes and R. Vergnioux, The boundary of universal discrete quantum groups, exactness and factoriality, Duke Math. J. 140 (2007), no. 1, 35–84.
  • R. Vergnioux, K-amenability for amalgamated free products of amenable discrete quantum groups, J. Funct. Anal. 212 (2004), no. 1, 206–221.
  • S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), 195–211.
  • S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613–665.
  • S. L. Woronowicz, Compact quantum groups, Symétries quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, pp. 845–884.