Noncommutative extensions of the Fourier transform and its logarithm
Volume 152 / 2002
Abstract
We introduce noncommutative extensions of the Fourier transform of probability measures and its logarithm to the algebra ${\cal A}(S)$ of complex-valued functions on the free semigroup $S={\rm FS}(\{z,w\})$ on two generators. First, to given probability measures $\mu$, $\nu$ with all moments finite, we associate states $\widehat{\mu}$, $\widehat{\nu}$ on the unital free *-bialgebra $({\cal B},\varepsilon ,{\mit\Delta})$ on two self-adjoint generators $X,X'$ and a projection $P$. Then we introduce and study cumulants which are additive under the convolution $\widehat{\mu}\star \widehat{\nu}=\widehat{\mu}\otimes \widehat{\nu} \circ {\mit\Delta}$ when restricted to the “noncommutative plane” ${\cal B}_{0}={\mathbb C}\langle X, X'\rangle$. We find a combinatorial formula for the Möbius function in the inversion formula and define the moment and cumulant generating functions, $M_{\widehat{\mu}}\{z,w\}$ and $L_{\widehat{\mu}}\{z,w\}$, respectively, as elements of ${\cal A}(S)$. When restricted to the subsemigroups ${\rm FS}(\{z\})$ and ${\rm FS}(\{w\})$, the function $L_{\widehat{\mu}}\{z,w\}$ coincides with the logarithm of the Fourier transform and with the $K$-transform of $\mu$, respectively. Moreover, $M_{\widehat{\mu}}\{z,w\}$ is a “semigroup interpolation” between the Fourier transform and the Cauchy transform of $\mu$. If one chooses a suitable weight function $W$ on the semigroup $S$, the moment and cumulant generating functions become elements of the Banach algebra $l^{1}(S,W)$.
