R-CYCLIC FAMILIES OF MATRICES IN FREE PROBABILITY

ALEXANDRU NICA DIMITRI SHLYAKHTENKO ROLAND SPEICHER

Abstract. We introduce the concept of “R-cyclic family” of matrices with entries in a non-commutative probability space; the definition consists in asking that only the “cyclic” non-crossing cumulants of the entries of the matrices are allowed to be non-zero.

Let A_{1}, …, A_{s} be an R-cyclic family of d × d matrices over a non-commutative probability space . We prove a convolution-type formula for the explicit computation of the joint distribution of A_{1}, …, A_{s} (considered in M_{d}(A) with the natural state), in terms of the joint distribution (considered in the original space ) 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 A_{1}, …, A_{s} be a family of d × d matrices over a non-commutative probability space , let D ⊂ M_{d}(A) denote the algebra of scalar diagonal matrices, and let C be the subalgebra of M_{d}(A) generated by {A_{1}, …, A_{s}} ∪ D. We prove that the R-cyclicity of A_{1}, …, A_{s} is equivalent to a property of C – namely that C is free from M_{d}(C), with amalgamation over D.