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RIESZ TRANSFORMS ON COMPACT QUANTUM GROUPS AND STRONG SOLIDITY

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  26 April 2021

Martijn Caspers Open the ORCID record for Martijn Caspers [Opens in a new window]
Martijn Caspers*
Affiliation:
TU Delft, EWI/DIAM, P.O.Box 5031, 2600GADelft, The Netherlands (m.p.t.caspers@tudelft.nl)
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Abstract

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One of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call ‘approximate linearity with almost commuting intertwiners’. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups.

We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient- ${\mathcal {S}}_2$ condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann–Ostrand property; in particular, the same strong solidity results follow again (now following [27]).

Keywords

quantum Markov semigroupscompact quantum groupsstrong solidityRiesz transforms

MSC classification

Secondary: 46L10: General theory of von Neumann algebras
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 6 , November 2022 , pp. 2135 - 2171
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Banica, T., Le groupe quantique compact libre $U(n)$ , Comm. Math. Phys. 190(1) 1997), 143–172.CrossRefGoogle Scholar
Banica, T., Bichon, J. and Collins, B., The hyperoctahedral quantum group, J. Ramanujan Math. Soc. 22(4) (2007), 345–384.Google Scholar
Banica, T. and Speicher, R., Liberation of orthogonal Lie groups, Adv. Math. 222(4) (2009) 1461–1501.CrossRefGoogle Scholar
Bichon, J., Free wreath product by the quantum permutation group, Algebr. Represent. Theory 7(4) (2004), 343–362.10.1023/B:ALGE.0000042148.97035.caCrossRefGoogle Scholar
Bichon, J., De Rijdt, A. and Vaes, A., Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups, Comm. Math. Phys. 262(3) (2006), 703–728.10.1007/s00220-005-1442-2CrossRefGoogle Scholar
Boutonnet, R., Houdayer, C. and Vaes, S., Strong solidity of free Araki-Woods factors, Amer. J. Math. 140(5) (2018), 1231–1252.CrossRefGoogle Scholar
Brannan, M., Approximation properties for free orthogonal and free unitary quantum groups, J. Reine Angew. Math. 672 (2012), 223–251.Google Scholar
Brannan, M., Reduced operator algebras of trace-preserving quantum automorphism groups, Doc. Math. 18 (2013), 1349–1402.Google Scholar
Brown, N. and Ozawa, N., C*-Algebras and Finite-Dimensional Approximations, Graduate Studies in Mathematics, 88 (American Mathematical Society, Providence, RI, 2008).CrossRefGoogle Scholar
Caspers, M., Gradient forms and strong solidity of free quantum groups, Math. Ann. 379(1–2) (2021), 271–324.CrossRefGoogle Scholar
Caspers, M. and Fima, P., Graph products of operator algebras, J. Noncommut. Geom. 11(1) (2017), 367–411.CrossRefGoogle Scholar
Caspers, M., Isono, Y. and Wasilewski, M., ${L}^2$ -cohomology, derivations and quantum Markov semi-groups on $q$ -Gaussian algebras, Int. Math. Res. Not. IMRN 2020, rnaa044.Google Scholar
Caspers, M. and Skalski, A., The Haagerup property for arbitrary von Neumann algebras, Int. Math. Res. Not. IMRN NNN(19) (2015), 9857–9887.10.1093/imrn/rnu252CrossRefGoogle Scholar
Caspers, M. and Skalski, A., The Haagerup approximation property for von Neumann algebras via quantum Markov semigroups and Dirichlet forms, Comm. Math. Phys. 336(3) (2015), 1637–1664.10.1007/s00220-015-2302-3CrossRefGoogle Scholar
Chifan, I., Sinclair, T. and Udrea, B., On the structural theory of II1 factors of negatively curved groups, II: Actions by product groups, Adv. Math. 245 (2013), 208–236.CrossRefGoogle Scholar
Cipriani, F., Franz, U. and Kula, A., Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory, J. Funct. Anal. 266(5) (2014), 2789–2844.10.1016/j.jfa.2013.11.026CrossRefGoogle Scholar
Cipriani, F. and Sauvageot, J. L., Derivations as square roots of Dirichlet forms, J. Funct. Anal. 201(1) (2003), 78–120.CrossRefGoogle Scholar
Cipriani, F. and Sauvageot, J. L., Amenability and subexponential spectral growth rate of Dirichlet forms on von Neumann algebras, Adv. Math. 322 (2017), 308–340.CrossRefGoogle Scholar
Daws, M., Operator biprojectivity of compact quantum groups, Proc. Amer. Math. Soc. 138(4) (2010), 1349–1359.10.1090/S0002-9939-09-10220-4CrossRefGoogle Scholar
Daws, M., Fima, P., Skalski, A. and White, S., The Haagerup property for locally compact quantum groups, J. Reine Angew. Math. 711 (2016), 189–229.Google Scholar
de Commer, K., Freslon, A. and Yamashita, M., CCAP for universal discrete quantum groups, Comm. Math. Phys. 331(2) (2014), 677–701. With an appendix by Stefaan Vaes.CrossRefGoogle Scholar
Effros, E. and Ruan, Z. J., Operator Spaces , London Mathematical Society Monographs, New Series, 23 (Clarendon Press, Oxford University Press, New York, 2000).Google Scholar
Fima, P. and Vergnioux, R., A cocycle in the adjoint representation of the orthogonal free quantum groups, Int. Math. Res. Not. IMRN 2015(20), 10069–10094.CrossRefGoogle Scholar
Freslon, A., Examples of weakly amenable discrete quantum groups, J. Funct. Anal. 265(9) (2013), 2164–2187.CrossRefGoogle Scholar
Freslon, A., On two-coloured noncrossing partition quantum groups, Trans. Amer. Math. Soc. 372(6) (2019), 4471–4508.CrossRefGoogle Scholar
Isono, Y., On bi-exactness of discrete quantum groups, Int. Math. Res. Not. IMRN NNN(11) (2015), 3619–3650.Google Scholar
Isono, Y., Examples of factors which have no Cartan subalgebras, Trans. Amer. Math. Soc. 367(11) (2015), 7917–7937.CrossRefGoogle Scholar
Jolissaint, P. and Martin, F., Algèbres de von Neumann finies ayant la propriété de Haagerup et semi-groupes L2-compacts, Bull. Belg. Math. Soc. Simon Stevin 11(1) (2004), 35–48.CrossRefGoogle Scholar
Junge, M., Neufang, M. and Ruan, Z.-J., A representation theorem for locally compact quantum groups, Internat. J. Math. 20(3) (2009), 377–400.CrossRefGoogle Scholar
Lemeux, F. and Tarrago, P., Free wreath product quantum groups: The monoidal category, approximation properties and free probability, J. Funct. Anal. 270(10) (2016), 3828–3883.CrossRefGoogle Scholar
Neshveyev, S. and Yamashita, M., Poisson boundaries of monoidal categories, Ann. Sci. Éc. Norm. Supér. (4) 50(4) (2017), 927–972.CrossRefGoogle Scholar
Okayasu, R. and Tomatsu, R., Haagerup approximation property for arbitrary von Neumann algebras, Publ. Res. Inst. Math. Sci. 51(3) (2015), 567–603.CrossRefGoogle Scholar
Ozawa, N., Solid von Neumann algebras, Acta Math. 192(1) (2004), 111–117.CrossRefGoogle Scholar
Ozawa, N. and Popa, S., On a class of II1 factors with at most one Cartan subalgebra, II, Amer. J. Math. 132(3) (2010), 841–866.CrossRefGoogle Scholar
Ozawa, N. and Popa, S., On a class of II1 factors with at most one Cartan subalgebra, Ann. of Math. (2) 172(1) (2010), 713–749.10.4007/annals.2010.172.713CrossRefGoogle Scholar
Peterson, J., ${L}_2$ -rigidity in von Neumann algebras, Invent. Math. 175(2) (2009), 417–433.CrossRefGoogle Scholar
Pisier, G., Introduction to Operator Space Theory , London Mathematical Society Lecture Note Series, 294 (Cambridge University Press, Cambridge, UK, 2003).CrossRefGoogle Scholar
Popa, S. and Vaes, S., Unique Cartan decomposition for II1 factors arising from arbitrary actions of hyperbolic groups, J. Reine Angew. Math. 694 (2014), 215–239.Google Scholar
Popa, S. and Vaes, S., Representation theory for subfactors, $\lambda$ -lattices and C*-tensor categories, Comm. Math. Phys. 340(3) (2015), 1239–1280.CrossRefGoogle Scholar
Raum, S., Isomorphisms and fusion rules of orthogonal free quantum groups and their free complexifications, Proc. Amer. Math. Soc. 140(9) (2012), 3207–3218.CrossRefGoogle Scholar
Raum, S. and Weber, M., Easy quantum groups and quantum subgroups of a semi-direct product quantum group, J. Noncommut. Geom. 9(4) (2015), 1261–1293.CrossRefGoogle Scholar
Takesaki, M., Theory of Operator Algebras. I. Encyclopaedia of Mathematical Sciences, 124, Operator Algebras and Non-commutative Geometry, 5 (Springer-Verlag, Berlin, 2002).10.1007/978-3-662-10453-8CrossRefGoogle Scholar
Takesaki, M., Theory of Operator Algebras. II (Springer-Verlag, Berlin, 2003).CrossRefGoogle Scholar
Tarrago, P. and Wahl, J., Free wreath product quantum groups and standard invariants of subfactors, Adv. Math. 331 (2018), 1–57.CrossRefGoogle Scholar
Tarrago, P. and Weber, M., The classification of tensor categories of two-colored noncrossing partitions, J. Combin. Theory Ser. A 154 (2018), 464–506.CrossRefGoogle Scholar
Vaes, S. and Vergnioux, R., The boundary of universal discrete quantum groups, exactness, and factoriality, Duke Math. J. 140(1) (2007), 35-84.CrossRefGoogle Scholar
Valvekens, M., ‘Some remarks on free products of rigid C*-2-categories’, Preprint, (2020), https://arxiv.org/abs/2008.12994.Google Scholar
Voiculescu, D., The analogues of entropy and of Fisher’s information measure in free probability theory. III. The absence of Cartan subalgebras, Geom. Funct. Anal. 6(1) (1996), 172–199.CrossRefGoogle Scholar
Wang, S., Free products of compact quantum groups, Comm. Math. Phys. 167(3) (1995), 671–692.CrossRefGoogle Scholar
Weber, M., On the classification of easy quantum groups, Adv. Math. 245 (2013), 500–533.CrossRefGoogle Scholar
Woronowicz, S. L., Compact matrix pseudogroups, Comm. Math. Phys. 111(4) (1987), 613–665.10.1007/BF01219077CrossRefGoogle Scholar
Woronowicz, S. L., Tannaka-Kreĭn duality for compact matrix pseudogroups: Twisted $\mathrm{SU}(N)$ groups, Invent. Math. 93(1) (1988), 35–76.10.1007/BF01393687CrossRefGoogle Scholar