Free Transportation Cost Inequalities for Noncommutative Multi-Variables

Abstract

Abstract. We prove the free analogue of the transportation cost inequality for tracial distributions of non-commutative self-adjoint (also unitary) multi-variables based on random matrix approximation procedure.

Notes on Microstate Free Entropy of Projections

“…The first inequality is shown in the same way as in [14,Lemma 1.3], while the second immediately follows from the inequality…”

“…With the above lemmas we can now prove the following transportation cost inequality in the essentially same manner as in [14].…”

“…Then one can choose a subsequence N 1 < N 2 < · · · so that Proof. The proof can be found in [14], even though only the self-adjoint and the unitary cases are treated there.…”

“…We note that such large deviation principle played quite an important role not only for the foundation of free entropy theory but also for getting free analogs of several probability theoretic inequalities (see [14] and the references therein). Recently, one more large deviation was shown in [12] for an independent pair of random projection matrices, including the explicit formula of the free entropy χ proj (p, q) of a projection pair (p, q).…”

Orbital Free Pressure and Its Legendre Transform

“…and letting k → ∞ we get the desired inequality, since the mean tracial state τ (0,A (k) ) N k converges to the free product tracial state ⋆ n i=1 τ i in the weak* topology as k → ∞ (see [11,Lemma 3.3]). The not necessarily unique (under τ ↾ CR(xi) = τ i , 1 ≤ i ≤ n) orbital equilibrium case can be reduced to the previous one with the help of a standard method based on e.g., [4, Lemma 6.2.43], see the final part of the proof of [14,Theorem 3.1].…”