Some estimates for non-microstates free entropy dimension with applications to q-semicircular families

By D. Shlyakhtenko. 2003

61 citations recorded by the source corpus.

We give a general estimate for the nonmicrostates free entropy dimension δ*(X 1 ,…X n ). If X 1 ,…X n generate a diffuse von Neumann algebra, we prove that δ*X 1 ,…X n ≥ 1. In the case that X 1 ,…,X n are q-semicircular variables as introduced by Bozejko and Speicher and q 2 n 1. We also show that for |q|<2−1, the von Neumann algebras generated by a finite family of q-Gaussian random variables satisfy a condition of Ozawa and are therefore solid: the relative commutant of any diffuse subalgebra must be hyperfinite. In particular, when these algebras are factors, they are prime and do not have property Γ.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/939477323a82bf22f8129dec515d2e943525770e

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Citations in the corpus

Examples of factors which have no Cartan subalgebras (2012, background): “See [28], [14], and [18] for other notions similar to condition (AO).”
A Kurosh-type theorem for type II1 factors (2004, background): “I would like to thank Nate Brown, Ken Dykema, Dimitri Shlyakhtenko and Georges Skandalis for many helpful discussions.”
L 2-homology for von Neumann algebras (2003, unknown): “,X σ n]). Theorem 4.8. One has dimM⊗¯Mo ∂ t F(B(L2(M))∩HSn HS ≤δ∗(X 1,...,Xn) δ⋆(X 1,...,Xn) ≤ dimM⊗¯Mo ∂ t F (B(L2(M)) w ∩HSn = ∆(F). Proof. The ﬁrst inequality is the statement of Corollary 2.12 in [Shl03b]. The second inequality is the statement of Theorem 4.4, together with the “dual” descriptio”