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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?)

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?

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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).

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