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The DT-operators are introduced, one for every pair (µ, c) consisting of a compactly supported Borel probability measure µ on the complex plane and a constant c > 0. These are operators on Hilbert space that are defined as limits in * -moments of certain upper triangular random matrices. The DT-operators include Voiculescu's circular operator and elliptic deformations of it, as well as the circular free Poisson operators. We show that every DT-operator is strongly decomposable. We also show that a DT-operator generates a II 1 -factor, whose isomorphism class depends only on the number and sizes of atoms of µ. Those DT-operators that are also R-diagonal are identified. For a quasi-nilpotent DT-operator T, we find the distribution of T * T and a recursion formula for general * -moments of T.

The algebra Mul [[B]] of formal multilinear function series over an algebra B and its quotient SymMul [[B]] are introduced, as well as corresponding operations of formal composition. In the setting of Mul [[B]], the unsymmetrized R-and T-transforms of random variables in B-valued noncommutative probability spaces are introduced. These satisfy properties analogous to the usual R-and T-transforms (the latter being just the reciprocal of the S-transform), but describe all moments of a random variable, not only the symmetric moments. The partially ordered set of noncrossing linked partitions is introduced and is used to prove properties of the unsymmetrized T-transform.

The interpolated free group factors L(¥ r ) for 1 < r < oo (also defined by F. Radulescu) are given another (but equivalent) definition as well as proofs of their properties with respect to compression by projections and free products. In order to prove the addition formula for free products, algebraic techniques are developed which allow us to show R*R*ί L(¥ 2 ) where R is the hyperfinite Hi -factor.Introduction. The free group factors L(F n ) for n -2, 3, ... , oo (introduced in [4]) have recently been extensively studied [11, 2, 5, 6, 7] using Voiculescu's theory of freeness in noncommutative probability spaces (see [8,9,10,11,12,13], especially the latter for an overview). One hopes to eventually be able to solve the isomorphism question, first raised by R. V. Kadison of whether L(F n ) = L(F m ) for n Φ m. In [7], F. Radulescu introduced Hi -factors L(F r ) for 1 < r < oo, equalling the free group factor L(¥ n ) when r = n e N\{0, 1} and satisfying

A random matrix model for freeness is extended and used to investigate free products of free group factors with matrix algebras and with the hyperfinite II 1 -factor. The latter is shown to be isomorphic to a free group factor having one additional generator.

We prove that the set of quantum correlations for a bipartite system of 5 inputs and 2 outputs is not closed. Our proof relies on computing the correlation functions of a graph, which is a concept that we introduce.2010 Mathematics Subject Classification. Primary 46L05; Secondary 47L90.