  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.
