  We solve two longstanding major problems in Free Probability. This is
achieved by generalising the theory to one with values in arbitrary commutative
algebras. We prove the existence of the multi-variable $S$-transform, and show
that it is naturally realised as a faithful linear representation. Further, we
prove that in dimension one, the analog of the classical relation between
addition and multiplication of independent random variables holds for free
random variables, if the co-domain is an algebra over the rationals. In this
case the multiplicative problem can be reduced to the additive one, which is
not true in dimensions greater than one. Finally, we classify the groups which
arise as joint distributions of $n$-tuples of non-commutative random variables,
endowed with the free convolution product, which is the binary operation that
encodes the multiplication of free $n$-tuples.
