Noncommutative martingale deviation and Poincaré type inequalities with applications

Abstract

Abstract. We prove a deviation inequality for noncommutative martingales by extending Oliveira's argument for random matrices. By integration we obtain a Burkholder type inequality with satisfactory constant. Using continuous time, we establish noncommutative Poincaré type inequalities for "nice" semigroups with a positive curvature condition. These results allow us to prove a general deviation inequality and a noncommutative transportation inequality due to Bobkov and Götze in the commutative case. To demonstrate our setting is general enough, we give various examples, including certain group von Neumann algebras, random matrices and classical diffusion processes, among others.

Subgaussian 1-cocycles on discrete groups

“…Recall Bakry-Émery's Γ 2 -criterion [3]: There exists α > 0 such that Γ ψ 2 (f, f ) αΓ ψ (f, f ) for all f ∈ LG for which both Γ ψ 2 (f, f ) and Γ ψ (f, f ) are well-defined. As observed in [26], in our context this condition is equivalent to the algebraic condition that Γ ψ 2 − αΓ ψ is a positive semidefinite form. The domain of Γ 2 is typically smaller than that of Γ.…”

“…In particular, choosing f (x) = x when d = 1 we recover the classical definition except for 1 p < 2. As a classical example, the Gaussian measure on R d satisfies (1.1) due to Pisier [35]; see [26] for another proof. More classical examples satisfying (1.1) can be found in [1] and the references therein.…”

“…Efraim and Lust-Piquard proved that (1.2) holds for Walsh systems and CAR algebras in [13]. In fact, we started to study the subgaussian behavior (1.2) of semigroups acting on a general noncommutative W * probability space (N , τ) in [26]. It was shown in [44] that the group measure space L ∞ (R d , γ d ) G satisfies (1.2), where the action and the gaussian measure γ d are associated to an orthogonal representation of G on a real Hilbert space, and the semigroup acting on L ∞ (R d , γ d ) G is a natural extension of the Ornstein-Uhlenbeck semigroup on L ∞ (R d , γ d ).…”

Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance

“…The proof is step-for-step the one from our previous paper, with only minor modifications, and we shall thereofre be brief. However, it is worth recording the more general result since this subject has recently attracted the attention of other researchers [42,62]. The connection between logarithmic Sobolev inequalities and transport inequalities of Talagrand type [68] was originally discovered and developed by Otto and Villani [55].…”

“…However, the gradient flow structure considered in Mielke's papers is in general different from the one introduced in the present paper, as the approach in [49,50] gives rise to nonlinear evolution equations that are different from the linear Lindblad equations that we obtain here. Also, Junge and Zeng [38] have recently developed an approach to some non-commutative functional inequalities involving a non-commutative analog of the 1-Wasserstein metric.…”