R-Cyclic Families of Matrices in Free Probability

✍ Scribed by Alexandru Nica; Dimitri Shlyakhtenko; Roland Speicher

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 293 KB
• Volume: 188
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2001.3814

✦ Synopsis

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 A 1 , ..., A s be an R-cyclic family of d × d matrices over a noncommutative probability space (A, j). 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 (A, j)) 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 noncommutative probability space (A, j), 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 } 2 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.