Noncommutative Martingales Research Articles

Mixed Lp$L^p$ estimates for transforms of noncommutative martingales

AbstractLet . The paper is devoted to the study of estimates for transforms of noncommutative martingales, under the assumption that the transforming sequence takes values in , . This goes beyond the usual context of and studied so far in the literature. The obtained constants are of optimal order at the endpoints, in addition the approach allows to obtain sharp values in the range . The proof rests on real interpolation‐type arguments for martingale transforms, which are of independent interest.

Noncommutative Nest Hardy Spaces and Martingales

Abstract Let $(\mathcal{M},\tau )$ be a finite von Neumann algebra equipped with a normalized faithful trace and let $\mathbb{A}$ be a nest of order type $\mathbb{N}$. The nest algebra is defined by $H_{\infty }^{r}(\mathbb{A})=\{x\in \mathcal{M}:ex=exe,e\in \mathbb{A}\}$, and the related noncommutative Hardy space $H_{p}^{r}(\mathbb{A})$ is the completion of $H_{\infty }^{r}(\mathbb{A})$ in $L_{p}(\mathcal{M})$. In the present paper, we make a connection between $H_{p}^{r}(\mathbb{A})$ and certain noncommutative martingale Hardy space via showing that triangular truncation is a concrete martingale transform. We include several related results for noncommutative nest Hardy spaces in the spirit of recent development of noncommutative martingale theory.

Failure of almost uniform convergence for noncommutative martingales

Failure of almost uniform convergence for noncommutative martingales

Complex interpolation between noncommutative martingale BMO spaces and Hardy–Orlicz spaces

Complex interpolation between noncommutative martingale BMO spaces and Hardy–Orlicz spaces

Average-Case Speedup for Product Formulas

Quantum simulation is a promising application of future quantum computers. Product formulas, or Trotterization, are the oldest and still remain an appealing method to simulate quantum systems. For an accurate product formula approximation, the state-of-the-art gate complexity depends on the number of terms in the Hamiltonian and a local energy estimate. In this work, we give evidence that product formulas, in practice, may work much better than expected. We prove that the Trotter error exhibits a qualitatively better scaling for the vast majority of input states, while the existing estimate is for the worst states. For general k-local Hamiltonians and higher-order product formulas, we obtain gate count estimates for input states drawn from any orthogonal basis. The gate complexity significantly improves over the worst case for systems with large connectivity. Our typical-case results generalize to Hamiltonians with Fermionic terms, with input states drawn from a fixed-particle number subspace, and with Gaussian coefficients (e.g., the SYK models). Technically, we employ a family of simple but versatile inequalities from non-commutative martingales called uniform smoothness, which leads to Hypercontractivity, namely p-norm estimates for k-local operators. This delivers concentration bounds via Markov’s inequality. For optimality, we give analytic and numerical examples that simultaneously match our typical-case estimates and the existing worst-case estimates. Therefore, our improvement is due to asking a qualitatively different question, and our results open doors to the study of quantum algorithms in the average case.

The sharp weighted maximal inequalities for noncommutative martingales

The purpose of the paper is to establish weighted maximal Lp-inequalities in the context of operator-valued martingales on semifinite von Neumann algebras. The main emphasis is put on the optimal dependence of the Lp constants on the characteristic of the weight involved. As applications, we establish weighted estimates for the noncommutative version of Hardy-Littlewood maximal operator and weighted bounds for noncommutative maximal truncations of a wide class of singular integrals.

Noncommutative martingale Hardy-Orlicz spaces: Dualities and inequalities

Noncommutative martingale Hardy-Orlicz spaces: Dualities and inequalities

Atomic decompositions for noncommutative martingales

We prove an atomic type decomposition for the noncommutative martingale Hardy space hp for all 0<p<2 by an explicit constructive method using algebraic atoms as building blocks. Using this elementary construction, we obtain a weak form of the atomic decomposition of hp for all 0<p<1, and provide a constructive proof of the atomic decomposition for p=1 which resolves a main problem on the subject left open for the last twelve years. We also study (p,∞)c-atoms, and show that every (p,2)c-atom can be decomposed into a sum of (p,∞)c-atoms; consequently, for every 0<p≤1, the (p,q)c-atoms lead to the same atomic space for all 2≤q≤∞. As applications, we obtain a characterization of the dual space of the noncommutative martingale Hardy space hp (0<p<1) as a noncommutative Lipschitz space via the weak form of the atomic decomposition. Our constructive method can also be applied to prove some sharp martingale inequalities.

Distributional inequalities for noncommutative martingales

We establish distributional estimates for noncommutative martingales, in the sense of decreasing rearrangements of the spectra of unbounded operators, which generalises the study of distributions of random variables. Our results include distributional versions of the noncommutative Stein, dual Doob, martingale transform and Burkholder-Gundy inequalities. Our proof relies upon new and powerful extrapolation theorems. As an application, we obtain some new martingale inequalities in symmetric quasi-Banach operator spaces and some interesting endpoint estimates. Our main approach demonstrates a method to build the noncommutative and classical probabilistic inequalities in an entirely operator theoretic way.

Interpolation and the John–Nirenberg inequality on symmetric spaces of noncommutative martingales

We prove various John–Nirenberg inequalities on symmetric spaces of noncommutative martingales, including the crude and fine versions, which extend the corresponding results of Junge and Musat (2007) and Hong and Mei (2012) in the $L_p$-case. As an applic

Interpolation between noncommutative martingale Hardy and BMO spaces: the case

AbstractLet$\mathcal {M}$be a semifinite von Nemann algebra equipped with an increasing filtration$(\mathcal {M}_n)_{n\geq 1}$of (semifinite) von Neumann subalgebras of$\mathcal {M}$. For$0&lt;p &lt;\infty $, let$\mathsf {h}_p^c(\mathcal {M})$denote the noncommutative column conditioned martingale Hardy space and$\mathsf {bmo}^c(\mathcal {M})$denote the column “little” martingale BMO space associated with the filtration$(\mathcal {M}_n)_{n\geq 1}$.We prove the following real interpolation identity: if$0&lt;p &lt;\infty $and$0&lt;\theta &lt;1$, then for$1/r=(1-\theta )/p$,$$ \begin{align*} \big(\mathsf{h}_p^c(\mathcal{M}), \mathsf{bmo}^c(\mathcal{M})\big)_{\theta, r}=\mathsf{h}_{r}^c(\mathcal{M}), \end{align*} $$with equivalent quasi norms.For the case of complex interpolation, we obtain that if$0&lt;p&lt;q&lt;\infty $and$0&lt;\theta &lt;1$, then for$1/r =(1-\theta )/p +\theta /q$,$$ \begin{align*} \big[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})\big]_{\theta}=\mathsf{h}_{r}^c(\mathcal{M}) \end{align*} $$with equivalent quasi norms.These extend previously known results from$p\geq 1$to the full range$0&lt;p&lt;\infty $. Other related spaces such as spaces of adapted sequences and Junge’s noncommutative conditioned$L_p$-spaces are also shown to form interpolation scale for the full range$0&lt;p&lt;\infty $when either the real method or the complex method is used. Our method of proof is based on a new algebraic atomic decomposition for Orlicz space version of Junge’s noncommutative conditioned$L_p$-spaces.We apply these results to derive various inequalities for martingales in noncommutative symmetric quasi-Banach spaces.

Sums of free variables in fully symmetric spaces

We give a method to obtain, from Voiculescu's inequality, norm estimates for sums of free variables with amalgamation in general fully symmetric spaces. We use these estimates to interpolate the Burkholder inequalities for non-commutative martingales. The method is also applicable to other similar settings. In that spirit, we improve known results on the non-commutative Johnson–Schechtman inequalities and recover Khinchin inequalities associated to free groups.

An equivalent quasinorm for the Lipschitz space of noncommutative martingales

Abstract In this paper, an equivalent quasinorm for the Lipschitz space of noncommutative martingales is presented. As an application, we obtain the duality theorem between the noncommutative martingale Hardy space h p c ( ℳ ) {h}_{p}^{c}( {\mathcal M} ) (resp. h p r ( ℳ ) {h}_{p}^{r}( {\mathcal M} ) ) and the Lipschitz space λ β c ( ℳ ) {\lambda }_{\beta }^{c}( {\mathcal M} ) (resp. λ β r ( ℳ ) {\lambda }_{\beta }^{r}( {\mathcal M} ) ) for 0 &lt; p &lt; 1 0\lt p\lt 1 , β = 1 p − 1 \beta =\tfrac{1}{p}-1 . We also prove some equivalent quasinorms for h p c ( ℳ ) {h}_{p}^{c}( {\mathcal M} ) and h p r ( ℳ ) {h}_{p}^{r}( {\mathcal M} ) for p = 1 p=1 or 2 &lt; p &lt; ∞ 2\lt p\lt \infty .

Square Functions for Noncommutative Differentially Subordinate Martingales

We prove inequalities involving noncommutative differentially subordinate martingales. More precisely, we prove that if $x$ is a self-adjoint noncommutative martingale and $y$ is weakly differentially subordinate to $x$ then $y$ admits a decomposition $dy=a +b +c$ (resp. $dy=z +w$) where $a$, $b$, and $c$ are adapted sequences (resp. $z$ and $w$ are martingale difference sequences) such that: $$ \Big\| (a_n)_{n\geq 1}\Big\|_{L_{1,\infty}({\mathcal M}\overline{\otimes}\ell_\infty)} +\Big\| \Big(\sum_{n\geq 1} \mathcal{E}_{n-1}|b_n|^2 \Big)^{{1}/{2}}\Big\|_{1, \infty} + \Big\| \Big(\sum_{n\geq 1} \mathcal{E}_{n-1}|c_n^*|^2 \Big)^{{1}/{2}}\Big\|_{1, \infty} \leq C\big\| x \big\|_1 $$ (resp. $$ \Big\| \Big(\sum_{n\geq1} |z_n|^2 \Big)^{{1}/{2}}\Big\|_{1, \infty} + \Big\| \Big(\sum_{n\geq 1} |w_n^*|^2 \Big)^{{1}/{2}}\Big\|_{1, \infty} \leq C\big\| x \big\|_1). $$ We also prove strong-type $(p,p)$ versions of the above weak-type results for $1<p<2$. In order to provide more insights into the interactions between noncommutative differential subordinations and martingale Hardy spaces when $1\leq p<2$, we also provide several martingale inequalities with sharp constants which are new and of independent interest. As a byproduct of our approach, we obtain new and constructive proofs of both the noncommutative Burkholder-Gundy inequalities and the noncommutative Burkholder/Rosenthal inequalities for $1<p<2$ with the optimal order of the constants when $p \to 1$.

Noncommutative Davis type decompositions and applications

We prove the noncommutative Davis decomposition for the column Hardy space $\H_p^c$ for all $0<p\leq 1$. A new feature of our Davis decomposition is a simultaneous control of $\H_1^c$ and $\H_q^c$ norms for any noncommutative martingale in $\H_1^c \cap \H_q^c$ when $q\geq 2$. As applications, we show that the Burkholder/Rosenthal inequality holds for bounded martingales in a noncommutative symmetric space associated with a function space $E$ that is either an interpolation of the couple $(L_p, L_2)$ for some $1<p<2$ or is an interpolation of the couple $(L_2, L_q)$ for some $2<q<\infty$. We also obtain the corresponding $\Phi$-moment Burkholder/Rosenthal inequality for Orlicz functions that are either $p$-convex and $2$-concave for some $1<p<2$ or are $2$-convex and $q$-concave for some $2<q<\infty$.

Noncommutative Burkholder/Rosenthal inequalities associated with convex functions

Nous prouvons des inégalités de martingales non commutatives associées à des fonctions convexes. Plus précisément, nous obtenons des analogues des inégalités de Burkholder non commutatives et des inégalités de Rosenthal non commutatives pour des $\Phi$-moments associés à toute fonction convexe $\Phi$ dont les indices de Matuzewska–Orlicz $p_{\Phi}$ et $q_{\Phi}$ vérifient $1<p_{\Phi}\leq q_{\Phi}<2$ ou $2<p_{\Phi}\leqq_{\Phi}<\infty$. Ces résultats généralisent les inégalités de Burkholder/Rosenthal non commutatives obtenues par Junge et Xu. L’ingrédient clé de notre approche est une version simultanée de l’inégalité de Burkholder récemment demontrée dans le cas des espaces $L_{p}$ non commutatifs pour $1<p<2$.

Duality for symmetric Hardy spaces of noncommutative martingales

We show the dual spaces of conditional Hardy space and symmetric Hardy space of noncommutative martingales. We derive relationship between the symmetric Hardy space of noncommutative martingales and its conditioned version.

Asymmetric Doob inequalities in continuous time

The present paper is devoted to the second part of our project on asymmetric maximal inequalities, where we consider martingales in continuous time. Let (M,τ) be a noncommutative probability space equipped with a continuous filtration of von Neumann subalgebras (Mt)0≤t≤1 whose union is weak-⁎ dense in M. Let Et denote the corresponding family of conditional expectations. As for discrete filtrations, we shall prove that for 1<p<2 and x∈Lp(M,τ) one can find a,b∈Lp(M,τ) and contractions ut,vt∈M such thatEt(x)=aut+vtbandmax⁡{‖a‖p,‖b‖p}≤cp‖x‖p. Moreover, aut and vtb converge in the row/column Hardy spaces Hpr(M) and Hpc(M) respectively. We also confirm in the continuous setting the validity of related asymmetric maximal inequalities which we recently found for discrete filtrations, including p=1. As for other results in noncommutative martingale theory, the passage from discrete to continuous index is quite technical and requires genuinely new methods. Our approach towards asymmetric maximal inequalities is based on certain construction of conditional expectations for a sequence of projective systems of Lp-modules. The convergence in Hpr(M) and Hpc(M) also imposes new algebraic atomic decompositions.

Noncommutative Valuation of Options

The aim of this note is to show that the classical results in finance theory for pricing of derivatives, given by making use of the replication principle , can be extended to the noncommutative world. We believe that this could be of interest in quantum probability. The main result called the First fundamental theorem of asset pricing , states that a noncommutative stock market admits no-arbitrage if and only if it admits a noncommutative equivalent martingale probability.

Johnson–Schechtman inequalities for noncommutative martingales

In this paper we study Johnson–Schechtman inequalities for noncommutative martingales. More precisely, disjointification inequalities of noncommutative martingale difference sequences are proved in an arbitrary symmetric operator space E(M) of a finite von Neumann algebra M without making any assumption on the Boyd indices of E. We show that we can obtain Johnson–Schechtman inequalities for arbitrary martingale difference sequences and that, in contrast with the classical case of independent random variables or the noncommutative case of freely independent random variables, the inequalities are one-sided except when E=L2(0,1). As an application, we partly resolve a problem stated by Randrianantoanina and Wu in [46]. We also show that we can obtain sharp Φ-moment analogues for Orlicz functions satisfying p-convexity and q-concavity for 1≤p≤2, q=2 and p=2, 2<q<∞. This is new even for the classical case. We also extend and strengthen the noncommutative Burkholder–Gundy inequalities in symmetric spaces and in the Φ-moment case.