Free Probability Theory studies a special class of 'noncommutative'random variables, which appear in the context of operators on Hilbert spaces and in one of the large random matrices. Since its emergence in the 1980s, free probability has evolved into an established field of mathematics with strong connections to other mathematical areas, such as operator algebras, classical probability theory, random matrices, combinatorics, representation theory of symmetric groups. Free probability also connects to more applied scientific fields, such as wireless communication in electrical engineering. This 2006 book gives a self-contained and comprehensive introduction to free probability theory which has its main focus on the combinatorial aspects. The volume is designed so that it can be used as a text for an introductory course (on an advanced undergraduate or beginning graduate level), and is also well-suited for the individual study of free probability.

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Let M denote the space of Borel probability measures on R. For every t ≥ 0 we consider the transformation B t : M → M defined bywhere ⊞ and ⊎ are the operations of free additive convolution and respectively of Boolean convolution on M, and where the convolution powers with respect to ⊞ and ⊎ are defined in the natural way. We show that B s • B t = B s+t , ∀ s, t ≥ 0 and that, quite surprisingly, every B t is a homomorphism for the operation of free multiplicativeWe prove that for t = 1 the transformation B 1 coincides with the canonical bijection B : M → M inf−div discovered by Bercovici and Pata in their study of the relations between infinite divisibility in free and in Boolean probability. Here M inf−div stands for the set of probability distributions in M which are infinitely divisible with respect to the operation ⊞. As a consequence, we have that B t (µ) is ⊞-infinitely divisible for every µ ∈ M and every t ≥ 1.On the other hand we put into evidence a relation between the transformations B t and the free Brownian motion; indeed, Theorem 4 of the paper gives an interpretation of the transformations B t as a way of re-casting the free Brownian motion, where the resulting process becomes multiplicative with respect to ⊠, and always reaches ⊞-infinite divisibility by the time t = 1.

Let D c (k) be the space of (non-commutative) distributions of k-tuples of selfadjoint elements in a C * -probability space. On D c (k) one has an operation of free additive convolution, and one can consider the subspace D inf-div c (k) of distributions which are infinitely divisible with respect to this operation. The linearizing transform for is the R-transform (one has R μ ν = R μ + R ν , ∀μ, ν ∈ D c (k)). We prove that, with M μ the moment series of μ. (The series η μ is the counterpart of R μ in the theory of Boolean convolution.) As a consequence, one can define a bijection B :We show that B is a multi-variable analogue of a bijection studied by Bercovici and Pata for k = 1, and we prove a theorem about convergence in moments which parallels the Bercovici-Pata result. On the other hand we prove the formulawith μ, ν considered in a space D alg (k) ⊇ D c (k) where the operation of free multiplicative convolution always makes sense. An equivalent reformulation of (II) is thatwhere is an operation on series previously studied by Nica and Speicher, and which describes the multiplication of free k-tuples in terms of their R-transforms. Formula (III) shows that, in a certain sense, η-series behave in the same way as R-transforms in connection to the operation of multiplication of free k-tuples of non-commutative random variables.

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