# Generalised operations in free harmonic analysis

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## Abstract

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.

## Keywords

Free probability Infinitely divisible distributions Semirings Witt vectors Positive definite functions## Notes

### Acknowledgements

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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