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\(R\)-diagonal pairs arising as free off-diagonal compression. (English) Zbl 0879.46031

Summary: We show that if one takes a free off-diagonal compression of a *-free family of elements in a tracial \(C^*\)-probability space, then one obtains a free family of \(R\)-diagonal pairs, in the sense of A. Nica and P. Speicher [Fields Inst. Commun. 12, 149-188 (1997)]. The proof relies on the matrix version of the full Fock space model for the \(R\)-transform, recently introduced by D. Shlyakhtenko [C. R. Acad. Sci., Paris, Ser. I 332, No. 7, 645-649 (1996; Zbl 0847.46031)].

MSC:

46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
47A20 Dilations, extensions, compressions of linear operators

Citations:

Zbl 0847.46031
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