The exponential map in non-commutative probability

TL;DR

This paper explores the wrapping transformation as a homomorphism connecting additive and multiplicative probability measures, extending it to non-commutative convolutions and revealing new structural insights.

Contribution

It demonstrates that the wrapping transformation preserves non-commutative convolutions and properties, providing simplified proofs of existing and new results.

Findings

W acts as a homomorphism between additive and multiplicative probability measures.

W preserves properties of measures and arrays across non-commutative convolutions.

New proofs for results related to multiplicative convolutions are established.

Abstract

The wrapping transformation $W$ is a homomorphism from the semigroup of probability measures on the real line, with the convolution operation, to the semigroup of probability measures on the circle, with the multiplicative convolution operation. We show that on a large class $\mathcal{L}$ of measures, $W$ also transforms the three non-commutative convolutions---free, Boolean, and monotone---to their multiplicative counterparts. Moreover, the restriction of $W$ to $\mathcal{L}$ preserves various qualitative properties of measures and triangular arrays. We use these facts to give short proofs of numerous known, and new, results about multiplicative convolutions.