On multiplicative conditionally free convolution

Abstract

Abstract. Using the combinatorics of non-crossing partitions, we construct a conditionally free analogue of Voiculescu's S-transform. The result is applied to the analytical description of conditionally free multiplicative convolution and the characterization of infinite divisibility.

A construction which relates c-freeness to infinitesimal freeness

“…One of the extensions is to the framework of c-free (shorthand for "conditionally free") independence, which was initiated in [6,7], and has accumulated a fairly large amount of work since then (see e.g. [5,16] and the references indicated there). Here one works with triples (A, ϕ, χ) where (A, ϕ) is an ncps (as above) and χ : A → C is an additional linear functional with χ(1 A ) = 1.…”

A Fock space model for addition and multiplication of c-free random variables

“…As also seen in [13], [8], this shift of coefficients is simplifying the notations in several recurrence relations from §2. For H a complex Hilbert space, we define T 0 (H) = H⊕(H⊗H)⊕(H⊗H⊗H)⊕.…”

“…(A more general approach to c-freeness, considering pairs of completely positive maps and conditional expectations have been pursued by F. Boca ([2]), K. Dykema and E. Blanchard ([6]), M. Popa, V. Vinnikov ( [9], [12]) etc). Addition of c-free random variables is studied in [4], where is constructed a c-free version of the R-transform, the c R-transform, with similar additivity and analytic properties (for φ = ψ, the two transforms coincide); multiplication of cfree random variables was studied in [13], where is constructed a c-free extension of the S-transform. In both cases, the proofs of the key properties (addition for the c R-and multiplication for the c S-transform) are combinatorial, much like the proofs from [8], heavily relaying on the properties on non-crossing partitions.…”

“…Section 2 presents basic definitions, the construction of the space E(H, T (K)) and an operator algebras model for c-free algebras. Section 3 presents the construction of some operators of prescribed c R and c S-transforms and Section 4 gives the proof for the additive, respective multiplicative properties of c R and c S. In this paper, rather than the S-or c S-transforms, we use, to simplify the notations their multiplicative inverses, the so-called T -and c T -transforms -i. e. T X (z) · S X (z) = 1 (see [5], [11]), respectively c T (z) · c S(z) = 1 (see [13]). …”

On fluctuations of traces of large matrices over a non-commutative algebra

“…This non-unital notion of non-commutative independence ( [16]) is by far less studied than freeness, but it was shown to be of relevance in some problems from Theoretical Physics ( [20]), Free Probability ( [13]), completely positive maps ( [14]) or Real Analysis ( [1]). We show that the Boolean cumulants of traces of ensembles of self-adjoint matrices with Bernoulli distributed boolean independent entries and constant matrices have a similar behavior to the classical cumulants of traces of Gaussian ensembles, as presented in [8], respectively to the free cumulants of traces of semicircular ensembles, as presented in Section 3.…”