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R-cyclic families of matrices in free probability
| Content Provider | Semantic Scholar |
|---|---|
| Author | Nica, Alexandru Shlyakhtenko, Dimitri |
| Copyright Year | 2000 |
| 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” non-crossing cumulants of the entries of the matrices are allowed to be non-zero. Let A1, . . . , As be an R-cyclic family of d×d matrices over a non-commutative 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 non-commutative 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. Research supported by a grant of NSERC, Canada. Partially supported by an NSF Postdoctoral Fellowship. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://arxiv.org/pdf/math/0101025v1.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Arabic numeral 0 Cobham's thesis Computation Convolution IBM Notes Nephrogenic Systemic Fibrosis |
| Content Type | Text |
| Resource Type | Article |
