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Communications in Mathematical Physics

, Volume 362, Issue 3, pp 761–799 | Cite as

Smooth Dense Subalgebras and Fourier Multipliers on Compact Quantum Groups

  • Rauan AkylzhanovEmail author
  • Shahn Majid
  • Michael Ruzhansky
Open Access
Article

Abstract

We define and study dense Frechet subalgebras of compact quantum groups realised as smooth domains associated with a Dirac type operator with compact resolvent. Further, we construct spectral triples on compact matrix quantum groups in terms of Clebsch–Gordon coefficients and the eigenvalues of the Dirac operator \({\mathcal{D}}\). Grotendieck’s theory of topological tensor products immediately yields a Schwartz kernel theorem for linear operators on compact quantum groups and allows us to introduce a natural class of pseudo-differential operators on them. It is also shown that regular pseudo-differential operators are closed under compositions. As a by-product, we develop elements of the distribution theory and corresponding Fourier analysis. We give applications of our construction to obtain sufficient conditions for LpLq boundedness of coinvariant linear operators. We provide necessary and sufficient conditions for algebraic differential calculi on Hopf subalgebras of compact quantum groups to extend to our proposed smooth subalgebra \({{C}^\infty_\mathcal {D}}\). We check explicitly that these conditions hold true on the quantum SU2q for both its 3-dimensional and 4-dimensional calculi.

References

  1. ANR15a.
    Akylzhanov, R., Nursultanov, E., Ruzhansky, M.: (2015) Hardy–Littlewood, Hausdorff–Young–Paley inequalities, and L pL q multipliers on compact homogeneous manifolds. arXiv:1504.07043, (2015)
  2. ANR15b.
    Akylzhanov K.R., Nursultanov D.E., Ruzhanskiĭ V.M.: Hardy–Littlewood–Paley-type inequalities on compact Lie groups. Mat. Zametki 100(2), 287–290 (2016)MathSciNetCrossRefGoogle Scholar
  3. ANR16.
    Akylzhanov R., Nursultanov E., Ruzhansky M.: Hardy–Littlewood–Paley inequalities and Fourier multipliers on SU(2). Studia Math. 234(1), 1–29 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. AR16.
    Akylzhanov R., Ruzhansky M.: Fourier multipliers and group von Neumann algebras. C. R. Math. Acad. Sci. Paris 354(8), 766–770 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. BC12.
    Bekjan N.T., Chen Z.: Interpolation and \({\Phi}\)-moment inequalities of noncommutative martin gales. Probab. Theory Relat. Fields 152(1), 179–206 (2012)CrossRefzbMATHGoogle Scholar
  6. BL76.
    Bergh, J., Löfström, J.: Interpolation spaces. An introduction. Springer-Verlag, Berlin-New York. Grundlehren der Mathematischen Wissenschaften, No. 223 (1976)Google Scholar
  7. BM15.
    Beggs E., Majid S.: Spectral triples from bimodule connections and Chern connections. J. Noncommut. Geom. 11(2), 669–701 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. CP08.
    Chakraborty S.P., Pal A.: Characterization of SUq (l + 1)-equivariant spectral triples for the odd dimensional quantum spheres. J. Reine Angew. Math. 623, 25–42 (2008)MathSciNetzbMATHGoogle Scholar
  9. CFK14.
    Cipriani F., Franz U., Kula A.: Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory. J. Funct. Anal. 266(5), 2789–2844 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. CL01.
    Connes A., Landi G.: Noncommutative manifolds, the instanton algebra and isospectral deformations. Comm. Math. Phys. 221(1), 141–159 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. Con95.
    Connes A.: Noncommutative geometry and reality. J. Math. Phys. 36(11), 6194–6231 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. Con96.
    Connes, A.: Gravity coupled with matter and the foundation of non-commutative geometry. Commun. Math. Phys. 182(1), 155–176 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. Con08.
    Connes, A., Moscovici, H.: Type III and spectral triples. In: Traces in Number Theory, Geometry and Quantum Fields, Aspects Math., E38, pp. 57–71. Friedr. Vieweg, Wiesbaden (2008)Google Scholar
  14. Con13.
    Connes A.: On the spectral characterization of manifolds. J. Noncommut. Geom. 7, 1–82 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Coo10.
    Cooney T.: A Hausdorff–Young inequality for locally compact quantum groups. Int. J. Math. 21(12), 1619–1632 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. DLS+05.
    Dabrowski, L., Landi, G., Sitarz, A., Suijlekom, W.van, Värilly, C.J.: The Dirac operator on SU q 2. Commun. Math. Phys. 259(3), 729–759 (2005)Google Scholar
  17. DK94.
    Dijkhuizen S.M., Koornwinder H.: \({\mathbb{C}_{q} [\mathfrak{g}]}\) algebras: a direct algebraic approach to compact quantum groups. Lett. Math. Phys. 32(4), 315–330 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. FGV00.
    Figueroa H., Gracia-Bondia J., Varilly J.: Elements of Noncommutative Geometry. Birkhäuser, New York (2000)zbMATHGoogle Scholar
  19. Fio98.
    Fiore G.: Deforming maps for Lie group covariant creation and annihilation operators. J. Math. Phys. 39(6), 3437–3452 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. Fol99.
    Folland, G.B.: Real analysis. Pure and Applied Mathematics (New York). John Wiley & Sons, Inc., New York, second edition. Modern techniques and their applications, A Wiley-Interscience Publication (1999)Google Scholar
  21. Gro55.
    Grothendieck A.: Produits tensoriels topologiques et espaces nucléaires. Mem. Amer. Math. Soc. 16, 140 (1955)zbMATHGoogle Scholar
  22. Haa79.
    Haagerup, U.: L p-spaces associated with an arbitrary von Neumann algebra. In Algèbres d’opérateurs et leurs applications en physique mathématique (Proc. Colloq., Marseille, 1977), volume 274 of Colloq. Int. CNRS, pp. 175–184. CNRS, Paris (1979)Google Scholar
  23. HL36.
    Hardy G., Littlewood J.: Some theorems concerning Fourier series and Fourier power series. Duke Math.J. 2, 354–382 (1936)MathSciNetCrossRefzbMATHGoogle Scholar
  24. HR74.
    Hewitt E., Ross A.K.: Rearrangements of L r Fourier series on compact Abelian groups. Proc. Lond. Math. Soc. 29(3), 317–330 (1974)CrossRefzbMATHGoogle Scholar
  25. JNR09.
    Junge M., Neufang M., Ruan Z.-J.: A representation theorem for locally compact quantum groups. Int. J. Math. 20(3), 377–400 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. KS12.
    Kaad J., Senior R.: A twisted spectral triple for quantum SU(2). J. Geom. Phys. 62(4), 731–739 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. Kos84.
    Kosaki H.: Applications of the complex interpolation method to a von Neumann algebra: non commutative L p-spaces. J. Funct. Anal. 56(1), 29–78 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  28. LNJP16.
    Lévy C., Neira Jiménez C., Paycha S.: The canonical trace and the noncommutative residue on the noncommutative torus. Trans. Amer. Math. Soc. 368(2), 1051–1095 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Lic90.
    Lichnerowicz A.: Spineurs harmoniques et spineurs-twisteurs en géométrie kählerienne et conformément kählerienne. C. R. Math. Acad. Sci. Paris. 311(13), 883–887 (1990)MathSciNetzbMATHGoogle Scholar
  30. LWW17.
    Liu Z., Wang S., Wu J.: Young’s inequality for locally compact quantum groups. J. Op. Theory 77(1), 109–131 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Maj95.
    Majid S.: Foundations of Quantum Group Theory. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  32. Maj03.
    Majid S.: Noncommutative Ricci curvature and Dirac operator on Cq [SL2] at roots of unity. Lett. Math. Phys. 63(1), 39–54 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. Maj15.
    Majid S.: Hodge star as braided Fourier transform. Algebr. Represent. Theory 20, 695–733 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  34. Maj16.
    Majid, S.: Noncommutative differential geometry. In: Bullet T.F.S., Smith, F. (eds) LTCC Lecture Notes Series: Analysis and Mathematical Physics, pp. 139–176. World Scientific, Singapore (2017)Google Scholar
  35. MMN+91.
    Masuda T., Mimachi K., Nakagami Y., Noumi M., Ueno K.: Representations of the quantum group SU q (2) and the little q-Jacobi polynomials. J. Funct. Anal. 99(2), 357–386 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  36. MVD98.
    Maes A., Van Daele A.: Notes on compact quantum groups. Nieuw Arch. Wisk. (4) 16(1-2), 73–112 (1998)MathSciNetzbMATHGoogle Scholar
  37. NT10.
    Neshveyev S., Tuset L.: The Dirac operator on compact quantum groups. J. Reine Angew. Math. 2010(641), 1–20 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  38. RT10.
    Ruzhansky M., Turunen V.: Pseudo-differential operators and symmetries, volume 2 of Pseudo- Differential Operators Theory and Applications. . Birkhäuser Verlag, Basel. (2010) Background analysis and advanced topics.CrossRefzbMATHGoogle Scholar
  39. RT13.
    Ruzhansky M., Turunen V.: Global quantization of pseudo-differential operators on compact Lie groups, SU(2), 3-sphere, and homogeneous spaces. Int. Math. Res. Not. IMRN. 11, 2439–2496 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  40. RT16.
    Ruzhansky M., Tokmagambetov N.: Nonharmonic analysis of boundary value problems. Int. Math. Res. Not. IMRN. 12, 3548–3615 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  41. TK86.
    Thierry F., Kosaki H.: Generalized s-numbers of \({\tau}\)-measurable operators. Pac. J. Math. 123(2), 269–300 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  42. Tre67.
    Trèves, F., Topological Vector Spaces, Distributions and Kernels. Academic Press, New York (1967)zbMATHGoogle Scholar
  43. Tri78.
    Triebel H.: Interpolation Theory, Function Spaces, Differential Operators, volume 18 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam (1978)Google Scholar
  44. Wor87.
    Woronowicz L.S.: Compact matrix pseudogroups. Comm. Math. Phys. 111(4), 613–665 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. Wor89.
    Woronowicz L.S.: Differential calculus on compact matrix pseudogroups (quantum groups). Comm. Math. Phys. 122(1), 125–170 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. Wor98.
    Woronowicz, L.S.: Compact quantum groups. In: Symétries Quantiques (Les Houches, 1995), pp. 845–884. North-Holland, Amsterdam (1998)Google Scholar
  47. Xu07.
    Xu, Q.: Operator spaces and noncommutative lp. The part on non-commutative Lp -spaces. Lectures in the Summer School on Banach spaces and Operator spaces, Nankai University-China (2007)Google Scholar
  48. You08.
    Youn S.-G.: Hardy–Littlewood inequalities on compact quantum groups of Kac type. Anal. PDE. 11(1), 237–261 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Imperial College LondonLondonUK

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