In traditional (commutative) probability theory, sums of random variables correspond to convolutions of distribution functions, which plays well with the Fourier Transform.


In *free* (noncommutative) probability theory, sums of random variables correspond to *free* convolutions of distribution functions, which plays well with the *R-Transform*. (See here for instance: [Relationship between R-transform and free convolution of random matrices?](https://mathoverflow.net/questions/76285/relationship-between-r-transform-and-free-convolution-of-random-matrices))




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Underlying the Fourier Transform is the representation theory of (locally compact) abelian groups.


**Question:** Is there representation theory underlying the $R$-Transform? Can the $R$-Transform be viewed as part of a theory of noncommutative harmonic analysis?



Relating the R-transform in free probability to noncommutative group representations
The $R$-transform is related to harmonic analysis around free products of groups.
Actually, the computational machinery for the $R$-transform was also found independently from Voiculescu and about the same time, by Woess, Cartwright and Soardi, and McLaughlin, in a more restricted setting of random walks on free product of groups; see, for example,
W.Woess: Nearest neighbour random walks on free products of discrete groups. Boll. Unione
Mat. Ital. VI. Ser. B 5, 961–982 (1986).


