Noncommutative Extensions of the Fourier Transform and Its Logarithm

We introduce noncommutative extensions of the Fourier transform of probability measures and its logarithm in the algebra A(S) of complex-valued functions on the free semigroup on two generators S = FS({z,w}). First, to given probability measures μ, ν whose all moments are finite, we associate states μ̂, ν̂ on the unital free *-bialgebra (B, ǫ,∆) on two self-adjoint generators X,X ′ and a projection P . Then we introduce and study cumulants which are additive under the convolution μ̂ ⋆ ν̂ = μ̂⊗ ν̂ ◦∆ when restricted to the “noncommutative plane” B0 = C〈X,X ′〉. We find a combinatorial formula for the Möbius function in the inversion formula and define the moment and cumulant generating functions, Mμ̂{z,w} and Lμ̂{z,w}, respectively, as elements of A(S). When restricted to the subsemigroups FS({z}) and FS({w}), the function Lμ̂{z,w} coincides with the logarithm of the Fourier transform and with the K-transform of μ, respectively. In turn, Mμ̂{z,w} is a “semigroup interpolation” between the Fourier transform and the Cauchy transform of μ. By choosing a suitable weight function W on the semigroup S, the moment and cumulant generating functions become elements of the Banach algebra l1(S,W ). Mathematics Subject Classification (2000): Primary 46L53, 60E10, 43A20; Secondary 06A07, 81R50