Skip to main content
SpringerLink
Log in

Positive Definite Functions on Coxeter Groups with Applications to Operator Spaces and Noncommutative Probability

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

A new class of positive definite functions related to colour-length function on arbitrary Coxeter group is introduced. Extensions of positive definite functions, called the Riesz–Coxeter product, from the Riesz product on the Rademacher (Abelian Coxeter) group to arbitrary Coxeter group is obtained. Applications to harmonic analysis, operator spaces and noncommutative probability are presented. Characterization of radial and colour-radial functions on dihedral groups and infinite permutation group are shown.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Belinschi S.T., Bożejko M., Lehner F., Speicher R.: The normal distribution is \({\boxplus}\)-infinitely divisible. Adv. Math. 226(4), 3677–3698 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bekka B., de la Harpe P., Valette A.: Kazhdan’s Property (T), vol. 11 of New Mathematical Monographs. Cambridge University Press, Cambridge (2008)

    Book  Google Scholar 

  3. Bożejko M., Ejsmont W., Hasebe T.: Fock space associated to Coxeter groups of type B. J. Funct. Anal. 269(6), 1769–1795 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bożejko M., Ejsmont W., Hasebe T.: Noncommutative probability of type D. Int. J. Math. 28(2), 1750010, 30 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bożejko M., Januszkiewicz T., Spatzier R.J.: Infinite Coxeter groups do not have Kazhdan’s property. J. Oper. Theory 19(1), 63–67 (1988)

    MATH  MathSciNet  Google Scholar 

  6. Brown, N.P., Ozawa, N.: C *-Algebras and Finite-Dimensional Approximations, vol. 88 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2008)

  7. Bourbaki, N.: Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines. Actualités Scientifiques et Industrielles, No. 1337. Hermann, Paris (1968)

  8. Bożejko M.: On \({\Lambda (p)}\) sets with minimal constant in discrete noncommutative groups. Proc. Am. Math. Soc. 51, 407–412 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bożejko M., Speicher R.: Interpolations between bosonic and fermionic relations given by generalized Brownian motions. Math. Z. 222(1), 135–159 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bożejko, M., Szwarc, R.: Algebraic length and Poincaré series on reflection groups with applications to representations theory. In: Asymptotic Combinatorics with Applications to Mathematical Physics (St. Petersburg, 2001), vol. 1815 of Lecture Notes in Mathematics, pp. 201–221, Springer, Berlin (2003)

  11. Buchholz A.: Norm of convolution by operator-valued functions on free groups. Proc. Am. Math. Soc. 127(6), 1671–1682 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. Buchholz A.: Optimal constants in Khintchine type inequalities for fermions, Rademachers and q-Gaussian operators. Bull. Pol. Acad. Sci. Math. 53(3), 315–321 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Bożejko M., Wysoczański J.: Remarks on t-transformations of measures and convolutions. Ann. Inst. Henri Poincaré Probab. Stat. 37(6), 737–761 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Bożejko, M, Yoshida, H.: Generalized q-deformed Gaussian random variables. In: Quantum Probability, vol. 73 of Banach Center Publication, pp. 127–140. Polish Acad. Sci. Inst. Math. Warsaw (2006)

  15. Cherix P.-A., Cowling M., Jolissaint P., Julg P., Valette A.: Groups with the Haagerup Property (Gromov’s a-T-Menability), vol. 197 of Progress in Mathematics. Birkhäuser Verlag, Basel (2001)

    Book  MATH  Google Scholar 

  16. De Cannière J., Haagerup U.: Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups. Am. J. Math. 107(2), 455–500 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  17. Eng O.D.: Quotients of Poincaré polynomials evaluated at −1. J. Algebr. Comb. 13(1), 29–40 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. Feller, W.: (1971) An Introduction to Probability Theory and Its Applications, vol, II. 2nd edition. Wiley, New York

  19. Fendler G.: A note on L-sets. Colloq. Math. 94(2), 281–284 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Fendler, G.: Weak amenability of Coxeter groups. arXiv:math/0203052 (2002)

  21. Haagerup, U.: The best constants in the Khintchine inequality. Stud. Math. 70(3), 231–283 (1982), 1981

  22. Haagerup, U.: An example of a nonnuclear C *-algebra, which has the metric approximation property. Invent. Math. 50(3), 279–293 (1978/1979)

  23. Hausdorff F.: Summationsmethoden und Momentfolgen. II. Math. Z. 9(3-4), 280–299 (1921)

    Article  MATH  MathSciNet  Google Scholar 

  24. Haagerup U., Pisier G.: Bounded linear operators between C *-algebras. Duke Math. J. 71(3), 889–925 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  25. Humphreys J.E.: Reflection Groups and Coxeter Groups, vol. 29 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  26. Januszkiewicz T.: For Coxeter groups z |g| is a coefficient of a uniformly bounded representation. Fund. Math. 174(1), 79–86 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  27. Köstler C., Speicher R.: A noncommutative de Finetti theorem: invariance under quantum permutations is equivalent to freeness with amalgamation. Commun. Math. Phys. 291(2), 473–490 (2009)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  28. Lehner F.: Cumulants in noncommutative probability theory. I. Noncommutative exchangeability systems. Math. Z. 248(1), 67–100 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  29. Lust-Piquard F.: Inégalités de Khintchine dans \({C_p\;(1 < p < \infty)}\). C. R. Acad. Sci. Paris Sér. I Math. 303(7), 289–292 (1986)

    MATH  MathSciNet  Google Scholar 

  30. Pisier G.: Introduction to Operator Space Theory, vol. 294 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  31. Reiner V.: Note on a theorem of Eng. Ann. Comb. 6(1), 117–118 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  32. Reiner V., Stanton D., White D.: The cyclic sieving phenomenon. J. Combin. Theory Ser. A 108(1), 17–50 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  33. Serre J.-P.: Cohomologie des groupes discrets. In: Prospects in Mathematics (Proceedings of Symposium, Princeton University, Princeton, 1970), pp. 77–169. Ann. of Math. Studies, No. 70. Princeton Univ. Press, Princeton (1971)

  34. Simon B.: Representations of Finite and Compact Groups, vol. 10 of Graduate Studies in Mathematics. American Mathematical Society, Providence (1996)

    Google Scholar 

  35. Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. A062991 (2001)

  36. Steinberg, R.: Endomorphisms of Linear Algebraic Groups. Memoirs of the American Mathematical Society, No. 80. American Mathematical Society, Providence (1968)

  37. Valette A.: Weak amenability of right-angled Coxeter groups. Proc. Am. Math. Soc. 119(4), 1331–1334 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  38. Yosida, K.: Functional Analysis, vol. 123 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 6th edition. Springer, Berlin (1980)

Download references

Author information

Authors and Affiliations

  1. Polska Akademia Nauk, ul. Kopernika 18, 50-001, Wrocław, Poland

    Marek Bożejko

  2. Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 48-300, Wrocław, Poland

    Światosław R. Gal & Wojciech Młotkowski

Authors
  1. Marek Bożejko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Światosław R. Gal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wojciech Młotkowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wojciech Młotkowski.

Additional information

Communicated by Y. Kawahigashi

Uffe Haagerup (1949–2015) in Memoriam.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bożejko, M., Gal, Ś.R. & Młotkowski, W. Positive Definite Functions on Coxeter Groups with Applications to Operator Spaces and Noncommutative Probability. Commun. Math. Phys. 361, 583–604 (2018). https://doi.org/10.1007/s00220-018-3160-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-018-3160-6

Search

Navigation