R-cyclic families of matrices in free probability
| dc.creator | Nica, Alexandru | |
| dc.creator | Shlyakhtenko, Dimitri | |
| dc.creator | Speicher, Roland | |
| dc.date | 2001-01-03 | |
| dc.date.accessioned | 2026-07-07T04:39:30Z | |
| dc.date.available | 2026-07-07T04:39:30Z | |
| dc.description | 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 \times 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 \times d matrices over a non-commutative probability space, let \D \subset M_{d} (\A) denote the algebra of scalar diagonal matrices, and let {\cal C} be the subalgebra of M_{d} (\A) generated by \{A_{1}, ..., A_{s} \} \cup \D. We prove that the R-cyclicity of A_{1}, ..., A_{s} is equivalent to a property of {\cal C} -- namely that {\cal C} is free from M_{d} (\C), with amalgamation over \D. | |
| dc.identifier | https://arxiv.org/abs/math/0101025 | |
| dc.identifier | http://arxiv.org/abs/math/0101025 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/60687 | |
| dc.subject | Operator Algebras | |
| dc.subject | Probability | |
| dc.title | R-cyclic families of matrices in free probability | |
| dc.type | text |