A new type of probability theory called “Free Probability Theory” has been developed mainly by Voiculescu for these fifteen years. This is nothing but a sort of non-commutative probability theory closely related to the free product of operator algebras, where the notion of freeness is essential instead of independence in classical probability theory. In the present book, the author develops a combinatorial theory of D. Voiculescu’s reduced free product [Lect. Notes Math. 1132, 556-588 (1985; Zbl 0618.46048)] with amalgamation by connecting this structure with the lattice of non-crossing partitions. Moreover, the corresponding operator-valued free probability theory can be expanded efficiently in terms of non-crossing cumulants. The book reveals how a theory of infinitely divisible distributions is developed effectively in this mathematical setting and also treats in particular Gaussian and compound Poisson distributions. The non-crossing cumulants can be used for the description of a projected dynamics by a new generalized master equation as well.
More precisely, Chapter I provides with preliminaries on non-crossing partitions. The material treated here is chiefly taken from the author’s previous work [Math. Ann. 298, No. 4, 611-628 (1994; Zbl 0791.06010)]. In Chapter II the concept of operator-valued multiplicative functions is presented. Chapter III is devoted to the first main theme, namely, amalgamated free products. Historically, the notion of this reduced free product was introduced by D. Voiculescu (1985) as a generalization of his scalar-valued free product, and the author [cf. op. cit.] realized that the lattice of non-crossing partitions governs the structure of the scalar-valued free product. Since a non-crossing partition \(\pi\in \text{NC}(n)\) corresponds canonically to a bracketing of a monomial of \(n\) factors, the combinatorial description admits a canonical generalization to the operator-valued case. In fact, two special types of multiplicative functions, such as the moment and the cumulant functions, play the key role in this chapter. The leading philosophy behind this combinatorial approach is that the free product is linearized by the cumulant functions.
The second main theme, operator-valued free probability theory is given in Chapter IV. We begin with regarding the elements \(a\) of an algebra \(A\) over \(B\) (= a fixed unital algebra) as \(B\)-valued random variables. This leads naturally to the operator-valued free analogues of many classical probabilistic concepts for instance to the notion of free convolution. The emphasis is principally put on the notion of cumulants, which linearize the free convolution. The basic ideas and fundamental definitions are greatly due to D. Voiculescu [Astérisque 232, 243-275 (1995; Zbl 0839.46060)]. Moreover, the author proposes a new concept of compound B-Poisson distributions (CBPD). As a matter of fact, given a random variable \(Y\), one can construct a CBPD \(\pi_Y\) by the prescription that the moments of \(Y\) give the cumulants of \(\pi_Y\) up to a common factor. Especially, §4.5 shows that infinitely divisible distributions can be approximated by compound B-Poisson distributions. On this account, each infinitely divisible distribution is realized by a special random variable on a full Fock space. Note that the notion of infinite divisibility has only a non-trivial meaning if we stay within the set of positive distributions. Therefore, \(B\) is always required to be a \(C^*\)-algebra all through this book.
Chapter V is devoted to applications of the theory to operator-valued stochastic processes and stochastic differential equations (SDEs). It is interesting to refer to the fact that some properties of distributions treated in the previous chapter have a more amenable formulation in terms of cumulants than in terms of moments. The most important of such properties is the product property which translates for the cumulants into a cluster property. The author points out that in the case of operator-valued SDEs the non-crossing cumulants are the only ones which behave nicely in all respects. Another peculiar feature for the theory consists in the point that for all \(B\)-valued stochastic processes one can construct Gaussian approximations. Indeed, what is more important is that positivity is preserved in making the Gaussian approximation.
As references, ninety-two papers are listed totally, including ten papers on the related works done by the author himself.