R-Cyclic Families of Matrices in Free Probability

Abstract

We introduce the concept of “R-cyclic family” of matrices with entries in a noncommutative probability space; the definition consists in asking that only the “cyclic” noncrossing cumulants of the entries of the matrices are allowed to be nonzero. Let A1, …, As be an R-cyclic family of d×d matrices over a noncommutative probability space (A, ϕ). We prove a convolution-type formula for the explicit computation of the joint distribution of A1, …, As (considered in Md(A) with the natural state), in terms of the joint distribution (considered in the original space (A, ϕ)) of the entries of the s matrices. Several important situations of families of matrices with tractable joint distributions arise by application of this formula. Moreover, let A1, …, As be a family of d×d matrices over a noncommutative probability space (A, ϕ), let D⊂Md(A) denote the algebra of scalar diagonal matrices, and let C be the subalgebra of Md(A) generated by {A1, …, As}∪D. We prove that the R-cyclicity of A1, …, As is equivalent to a property of C—namely that C is free from Md(C), with amalgamation over D.