Generalised operations in free harmonic analysis


In this article we study the formal side of operations in free harmonic analysis and examine the emerging general picture of all this. We establish an analytic correspondence of semi-rings between Witt vectors and free probability, by building on previous joint work with Friedrich and McKay (Formal groups, Witt vectors and free probability, 2012. arXiv:1204.6522). In particular, an exponential map, which relates the free additive convolution semigroup on \({\mathbb {R}}\) with the free multiplicative convolution semigroup on either the unit circle or the positive real axis of compactly supported, freely infinitely divisible probability measures, is derived with complex analytic methods. Then we define several novel operations on these sets, discuss their relation with classically infinitely divisible measures and determine the internal geometry of the spaces involved. Finally, we formalise the structure induced by the various operations we have introduced, in the language of operads and algebraic theories.

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  1. 1.

    Anshelevich, M., Arizmendi, O.: The exponential map in non-commutative probability. Int Math Res Not. 17, 5302–5342 (2016).

    Article  MATH  Google Scholar 

  2. 2.

    Barndorff-Nielsen, O., Thorbørnsen, S.: Classical and Free Infinite Divisibility and Lévy Processes, Quantum Independent Increment Processes II’, LNM, vol. 1866, pp. 33–159. Springer, Berlin (2005)

    Google Scholar 

  3. 3.

    Bercovici, H., Pata, V., Biane, Ph: Stable laws and domains of attraction in free probability theory. Ann. Math. 149(3), 1023–1060 (1999)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bercovici, H., Voiculescu, D.: Lévy–Hinc̆in type theorems for multiplicative and additive free convolution. Pac. J. Math. 153(2), 217–248 (1992)

    Article  Google Scholar 

  5. 5.

    Bercovici, H., Voiculescu, D.: Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42, 733–773 (1993)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Berstel, J., Perrin, D.: Theory of Codes, Pure and Applied Mathematics. Academic Press, Cambridge (1985)

    Google Scholar 

  7. 7.

    Cébron, G.: Matricial model for the free multiplicative convolution. Ann. Probab. 44(4), 2427–2478 (2016)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Friedrich, R., McKay, J.: Free probability theory and complex cobordism, C. R. Math. Rep. Acad. Sci. Canada 33(4), 116–122 (2011)

    MATH  Google Scholar 

  9. 9.

    Friedrich, R., McKay, J.: Formal Groups, Witt Vectors and Free Probability (2012). arXiv:1204.6522

  10. 10.

    Friedrich, R., McKay, J.: The \(S\)-Transform in Arbitrary Dimensions (2013). arXiv:1308.0733

  11. 11.

    Friedrich, R., McKay, J.: Almost Commutative Probability Theory (2013). arXiv:1309.6194

  12. 12.

    Friedrich, R.: (Co)monads in Free Probability Theory (2017). arXiv:1709.02973

  13. 13.

    Fritz, T.: Convex Spaces I: Definition and Examples (2009). arXiv:0903.5522v3

  14. 14.

    Hazewinkel, M.: Witt vectors. Part 1, revised version: 20 April (2008)

  15. 15.

    Jacobs, B.: Convexity, duality and effects, theoretical computer science: 6th IFIP TC 1/WG 2.2 International Conference, TCS 2010, Held as Part of WCC 2010, Brisbane, Australia, September 20–23, 2010. Proceedings. Springer, p. 1–19 (2010)

  16. 16.

    Klenke, A.: Probability Theory: A Comprehensive Course, 2nd edn. Springer, Berlin (2013)

    Google Scholar 

  17. 17.

    Leinster, T.: An Operadic Introduction to Entropy’, \(n\)-Category Café blog post (2011). Accessed 8 Jan 2019

  18. 18.

    Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer, Berlin (1997)

    Google Scholar 

  19. 19.

    Male, C.: The Distributions of Traffics of Large Random Matrices and Their Free Product (2011). arXiv:1111.4662

  20. 20.

    Mastnak, M., Nica, A.: Hopf algebras and the logarithm of the \(S\)-transform in free probability. Trans. AMS 362, 3705–3743 (2010)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability, LMS LNS, vol. 335. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  22. 22.

    Parthasarathy, K.R.: Probability Measures on Metric Spaces, vol. 352. AMS Chelsea Publishing, Madison (2005)

    Google Scholar 

  23. 23.

    Speicher, R.: Free probability theory and non-crossing partitions. Séminaire Lotharingien de Combinatoire, B39c, 38 (1997)

  24. 24.

    Voiculescu, D.V.: Addition of certain non-commuting random variables. J. Funct. Anal. 66, 323–335 (1986)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Voiculescu, D.V.: Multiplication of certain non-commuting random variables. J. Oper. Theory 18, 223–235 (1987)

    MATH  Google Scholar 

  26. 26.

    Voiculescu, D.V., Dykema, K.J., Nica, A.: Free Random Variables: A Noncommutative Probability Approach to Free Products with Applications to Random Matrices, Operator Algebras, and Harmonic Analysis on Free Groups. AMS (1992)

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The author thanks: John McKay for the discussions and his continuous interest. Guillaume Cébron for the previous discussions and the helpful technical comments he made at various occasions. Roland Speicher for numerous discussions, his comments and his support. Dan Voiculescu for his comment on a previous version. Finally, he thanks the anonymous referee for the comments and suggestions which helped to improve this article, and the MPI in Bonn for its hospitality. The author is supported by the ERC advanced grant “Noncommutative distributions in free probability”.

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Correspondence to Roland M. Friedrich.

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Friedrich, R.M. Generalised operations in free harmonic analysis. Semigroup Forum 99, 632–654 (2019).

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  • Free probability
  • Infinitely divisible distributions
  • Semirings
  • Witt vectors
  • Positive definite functions