Noncommutative extensions of the Fourier transform and its logarithm

Abstract

We introduce and study noncommutative extensions of the Fourier transform and its logarithm to the algebra of functions on the free semigroup FS(2) on two generators with the convolution multiplication. These extensions are new types of moment and cumulant generating functions, respectively, the latter corresponding to the cumulants which are additive under the so-called filtered convolution on the free *-algebra on two generators. This algebra plays the role of a ``noncommutative plane'' built on the ``classical real line'' and the ``boolean real line''. The restrictions of the cumulant generating function to the commutative subsemigroups generated by single generators give the logarithm of the Fourier transform and the K-transform in the boolean case, respectively. In turn, the moment generating function is a ``semigroup interpolation'' between the Fourier transform and the Cauchy transform. Using suitable weight function $W$ on the semigroup, both generating functions become elements of the Banach algebra $l^{1}(FS(2),W)$. The main results of the paper are based on the new combinatorics developed for the cumulants, the Moebius function and the moment-cumulant formulas.