In the process of developing the theory of free probability and free entropy, Voiculescu introduced in 1991 a random matrix model for a free semicircular system. Since then, random matrices have played a key role in von Neumann algebra theory (cf. [V8], [V9]). The main result of this paper is the following extension of Voiculescu's random matrix result: Let (Xr ) be a system of r stochastically independent n × n Gaussian self-adjoint random matrices as in Voiculescu's random matrix paper [V4], and let (x 1 , . . . , x r ) be a semi-circular system in a C * -probability space. Then for every polynomial p in r noncommuting variablesfor almost all ω in the underlying probability space. We use the result to show that the Ext-invariant for the reduced C * -algebra of the free group on 2 generators is not a group but only a semi-group. This problem has been open since Anderson in 1978 found the first example of a C * -algebra A for which Ext(A) is not a group.

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This is the continuation of a previous article that studied the relationship between the classes of infinitely divisible probability measures in classical and free probability, respectively, via the Bercovici-Pata bijection. Drawing on the results of the preceding article, the present paper outlines recent developments in the theory of Lé vy processes in free probability.T he present article continues the account, begun in ref. 1, of some recent developments in the theory of free infinite divisibility and free Lévy processes. Lévy processes are stochastic processes with independent and stationary increments. In general, processes with ''independent'' (but not necessarily stationary) increments, where ''independent'' can have a variety of meanings, are objects of wide current interest in stochastics (i.e., probability and statistics together) and mathematical physics. Not only are processes of this type of great interest in themselves, but they occur as important building blocks in other more structured processes. Ref. The present article outlines several results on processes with freely independent (and stationary) increments that are closely parallel to key results from the theory of Lévy processes in classical probability theory. In light of these and other related findings it seems certain that much more of interest can be done in studying the similarities, as well as the intriguing differences, between processes based on classical stochastic independence and free independence, respectively.In Lévy Processes in Free Probability, we define free Lévy processes in complete analogy with the definition of classical Lévy processes. We show, subsequently, how the Bercovici-Pata bijection ⌳ studied in ref. 1 gives rise to a one-to-one (in law) correspondence between classical and free Lévy processes by virtue of its algebraic and topological properties, which were presented in ref.1. In Self-Decomposability and Free Stochastic Integration we use the properties of ⌳ to construct certain stochastic integrals with respect to (w.r.t.) free Lévy processes, and we derive the free counterpart of the well known integral representation of classically self-decomposable random variables. We describe in The Lévy-Itô Decomposition a free version of the key Lévy-Itô decomposition of classical Lévy processes. Finally, in Further Connections Between the Classical and Free Cases a stochastic interpretation of the Bercovici-Pata bijection is given.Throughout the present article, we make use, often without further comments, of the notations, definitions, etc. that were introduced in the preceding article (1). Lé vy Processes in Free ProbabilityIn classical probability, Lévy processes form a very important area of research, both from the theoretical and applied points of view (see refs. 2 and 6-9). In free probability, such processes have already received quite a lot of attention (e.g. see refs. 10-12).1.1. Definition: A free Lévy process (in law), affiliated with a W*-probability space (A, ), is a family (Z t ) tՆ0 of self-adjoint oper...