Avitzour [Avi] and Voiculescu [Voil] introduced around 1982, following earlier
investigations of Ching [Chi], the notion of a reduced free product of C*-algebras.
This construction was meant as a replacement of the concept 'tensor product' in
order to get another possibility for constructing new C*-algebras from given ones.
The free product was modelled according to the familiar notion of a free product
of groups. It was Voiculescu who realized that this concept of a free product of
C*-algebras is not only important as a technical tool in the structure theory of
C*-algebras, but that it deserves attention on its own. In particular, he recognized
that the concept of freeness is quite analogous to the classical probabilistic notion
of independence and thus leads canonically to the notion of 'free convolution'. In a
series of papers [Voi2,Voi3,Voi5,BVl,BV2] (see also [VDN]) he developed the free
analogues of many classical probabilistic concepts. The main tool for his investiga-
tions on the free convolution is the 'i?-transform', which replaces the logarithm of
the classical Fourier transform.
In our own work we tried to understand the results of Voiculescu from a more
combinatorial point of view, in the tradition of the algebraic approach of von
Waldenfels [GvW] to central limit theorems. In the course of our investigations it
turned out that the structure of the free convolution and, more generally, of the free
product is governed by the lattice of non-crossing partitions. These non-crossing
partitions were introduced in 1972 by Kreweras [Kre], but up to our investigations
they were only examined from a purely combinatorial point of view and no connec-
tion with probabilistic notions has been made. The first appearing of non-crossing
partitions in the quantum probabilistic context was in our proof of the free cen-
tral and Poisson limit theorems [Spel]. Inspired by Rota's combinatorial point of
view on classical probability theory [Rotl,Rot2], we could finally describe the free
convolution and the free product in terms of multiplicative functions on the lat-
tice of non-crossing partitions [Spe4]. This again shows the complete analogy of
freeness with independence, since the latter can be described in the same way by
multiplicative functions on the lattice of all partitions.
Our work centers around the notion of 'non-crossing' cumulants, which are calcu-
lated in a very specific way from moments with the help of the lattice of non-crossing
partitions. These cumulants have the characterizing property that they linearize
the free convolution. Abstractly, these quantities appeared also in Voiculescu's in-
vestigations on the free convolution, but only their concrete description with the
help of non-crossing partitions clarifies their structure.
Our approach to the free convolution and free product gives a complementary
point of view of the more operator algebraic investigations of Voiculescu. In par-
ticular, the structure of the i?-transform becomes more transparent and the main
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