arXiv 0810.57052008-10math.OAmath.FA

Remarks on the non-commutative Khintchine inequalities for $0<p<2$

Gilles Pisier

备注：: Some more minor corrections
发表信息：: J. Funct. Anal. 256 (2009), no. 12, 4128--4161

英文摘要：

We show that the validity of the non-commutative Khintchine inequality for some $q$ with $1<q<2$ implies its validity (with another constant) for all $1\le p<q$. We prove this for the inequality involving the Rademacher functions, but also for more general "lacunary" sequences, or even non-commutative analogues of the Rademacher functions. For instance, we may apply it to the "Z(2)-sequences" previously considered by Harcharras. The result appears to be new in that case. It implies that the space $\ell^n_1$ contains (as an operator space) a large subspace uniformly isomorphic (as an operator space) to $R_k+C_k$ with $k\sim n^{\frac12}$. This naturally raises several interesting questions concerning the best possible such $k$. Unfortunately we cannot settle the validity of the non-commutative Khintchine inequality for $0<p<1$ but we can prove several would be corollaries. For instance, given an infinite scalar matrix $[x_{ij}]$, we give a necessary and sufficient condition for $[\pm x_{ij}]$ to be in the Schatten class $S_p$ for almost all (independent) choices of signs $\pm 1$. We also characterize the bounded Schur multipliers from $S_2$ to $S_p$. The latter two characterizations extend to $0<p<1$ results already known for $1\le p\le2$. In addition, we observe that the hypercontractive inequalities, proved by Carlen and Lieb for the Fermionic case, remain valid for operator space valued functions, and hence the Kahane inequalities are valid in this setting.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

The non-commutative Khintchine inequalities for p<1

We give a proof of the Khintchine inequalities in non-commutative $L_p$-spaces for all $0< p<1$. These new inequalities are valid for the Rademacher functions or Gaussian random variables, but also for more general sequences, e.g. for the analogues of such random variables in free probability. We also prove a factorization for operators from a Hilbert space to a non commutative $L_p$-space, which is new for $0<p<1$. We end by showing that Mazur maps are Hölder on semifinite von Neumann algebras. The main tool is a new form of Hölder inequality for non commutative Lp spaces with weights.

2016-01-06math.FAmath.OA

Some remarks on noncommutative Khintchine inequalities

Normalized free semi-circular random variables satisfy an upper Khintchine inequality in $L_\infty$. We show that this implies the corresponding upper Khintchine inequality in any noncommutative Banach function space. As applications, we obtain a very simple proof of a well-known interpolation result for row and column operator spaces and, moreover, answer an open question on noncommutative moment inequalities concerning a paper by Bekjan and Chen.

2011-08-26math.FAmath.OA

辛钦不等式、迹幺半群与Turán型问题

Khintchine inequalities, trace monoids and Turán-type problems

本文通过迹幺半群与凯莱图谱半径刻画自由和张量独立半圆变量混合和的范数，建立非交换概率、组合群论与极值图论的联系，并提出Turán型极值问题。

2025-06-03math.COmath.FAmath.OA

Non-commutative Khintchine type inequalities associated with free groups

Let $\Free_n$ denote the free group with $n$ generators $g_1, g_2, ..., g_n$. Let $λ$ stand for the left regular representation of $\Free_n$ and let $τ$ be the standard trace associated to $λ$. Given any positive integer $d$, we study the operator space structure of the subspace $\Word_p(n,d)$ of $L_p(τ)$ generated by the family of operators $λ(g_{i_1}g_{i_2} ... g_{i_d})$ with $1 \le i_k \le n$. Moreover, our description of this operator space holds up to a constant which does not depend on $n$ or $p$, so that our result remains valid for infinitely many generators. We also consider the subsp…

2005-08-26math.FAmath.OA

Sum 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 Khintchine inequalities associated to free groups.

2020-11-19math.FAmath.OA

Multi-indexed p-orthogonal sums in non-commutative Lebesgue spaces

In this paper we extend a recent Pisier's inequality for p-orthogonal sums in non-commutative Lebesgue spaces. To that purpose, we generalize the notion of p-orthogonality to the class of multi-indexed families of operators. This kind of families appear naturally in certain non-commutative Khintchine type inequalities associated with free groups. Other p-orthogonal families are given by the homogeneous operator-valued polynomials in the Rademacher variables or the multi-indexed martingale difference sequences. As in Pisier's result, our tools are mainly combinatorial.

2004-04-15math.COmath.OA

更多相关论文收起更多论文

Recurrence and ergodicity in unital *-algebras

Results concerning recurrence and ergodicity are proved in an abstract Hilbert space setting based on the proof of Khintchine's recurrence theorem for sets, and on the Hilbert space characterization of ergodicity. These results are carried over to a non-commutative *-algebraic setting using the GNS-construction. This generalizes the corresponding measure theoretic results, in particular a variation of Khintchine's Theorem for ergodic systems, where the image of one set overlaps with another set, instead of with itself.

2002-08-10math.DSmath.OA

Real interpolation between row and column spaces

We give an equivalent expression for the $K$-functional associated to the pair of operator spaces $(R,C)$ formed by the rows and columns respectively. This yields a description of the real interpolation spaces for the pair $(M_n(R), M_n(C))$ (uniformly over $n$). More generally, the same result is valid when $M_n$ (or $B(\ell_2)$) is replaced by any semi-finite von Neumann algebra. We prove a version of the non-commutative Khintchine inequalities (originally due to Lust--Piquard) that is valid for the Lorentz spaces $L_{p,q}(τ)$ associated to a non-commutative measure $τ$, simultaneously for t…

2011-09-08math.FAmath.OA

Fock空间上的矩阵浓度与等价算子

Matrix Concentration and Equivalent Operators on Fock Spaces

ETH Zürich（苏黎世联邦理工学院）；Cheriton School of Computer Science, University of Waterloo（滑铁卢大学切里顿计算机科学学院）；Johns Hopkins University（约翰斯·霍普金斯大学）

本文提出一种通过Fock空间上确定性算子范数控制来证明矩阵浓度不等式的方法，获得非交换Khintchine不等式的强化结果，并统一处理多种非交换随机变量模型。

2026-10-02math.OAmath.PR

Martingale inequalities and Operator space structures on $L_p$

We describe a new operator space structure on $L_p$ when $p$ is an even integer and compare it with the one introduced in our previous work using complex interpolation. For the new structure, the Khintchine inequalities and Burkholder's martingale inequalities have a very natural form:\ the span of the Rademacher functions is completely isomorphic to the operator Hilbert space $OH$, and the square function of a martingale difference sequence $d_n$ is $Σ\ d_n\otimes \bar d_n$. Various inequalities from harmonic analysis are also considered in the same operator valued framework. Moreover, the ne…

2013-07-19math.FAmath.OAmath.PR

关于非交换 $L^p$ 空间的一点注记

A remark on non-commutative $L^p$-spaces

本文显式刻画了张量积 von Neumann 代数对应的 Haagerup 与 Kosaki 非交换 $L^p$ 空间，并说明其在算子代数量子信息理论中的用途。

2023-07-20math-phmath.MPmath.OA

Some remarks on derivations on the algebra of operators in Hilbert pro-C*-bimodules

Suppose $A$ is a pro-C*-algebra. Let $L_{A}(E)$ be the pro-C*-algebra of adjointable operators on a Hilbert $A$-module $E$ and let $K_{A}(E)$ be the closed two sided $*$-ideal of all compact operators on $E$. We prove that if $E$ be a full Hilbert $A$-module, the innerness of derivations on $K_{A}(E)$ implies the innerness of derivations on $L_{A}(E)$. We show that if $A$ is a commutative pro-C*-algebra and $E$ is a Hilbert $A$-bimodule then every derivation on $K_{A}(E)$ is zero. Moreover, if $A$ is a commutative $σ$-C*-algebra and $E$ is a Hilbert $A$-bimodule then every derivation on $L_{A}…

2016-12-10math.OA

Remarks on Morphisms of Spectral Geometries

Having in view the study of a version of Gel'fand-Neumark duality adapted to the context of Alain Connes' spectral triples, in this very preliminary review, we first present a description of the relevant categories of geometrical spaces, namely compact Hausdorff smooth finite-dimensional orientable Riemannian manifolds (or more generally Hermitian bundles of Clifford modules over them); we give some tentative definitions of the relevant categories of algebraic structures, namely "propagators" and "spectral correspondences" of commutative Riemannian spectral triples; and we provide a constructi…

2014-09-04math.CTmath.OA

关于单位球上双全纯映射的量子注记

A quantum remark on biholomorphic mappings on the unit ball

本文研究带算子值期望的非交换概率论，证明酉随机变量单调增量过程分布的矩生成函数能在特定高维单位球上生成双全纯映射。

2024-02-18math.CVmath.OA

Strong convergence for reduced free products (Remarks on a result by Paul Skoufranis)

Using an inequality due to Ricard and Xu, we give a different proof of Paul Skoufranis's recent result showing that the strong convergence of possibly non-commutative random variables $X^{(k)}\to X$ is stable under reduced free product with a fixed non-commutative random variable $Y$. In fact we obtain a more general fact: assuming that the families $X^{(k)}=\{X_i^{(k)}\}$ and $Y^{(k)}=\{Y_j^{(k)}\}$ are $*$-free as well as their limits (in moments) $X =\{X_i \}$ and $Y =\{Y_j\}$, the strong convergences $X^{(k)}\to X$ and $Y^{(k)}\to Y$ imply that of $\{X^{(k)},Y^{(k)} \}$ to $\{X ,Y\}$. Phra…

2016-02-25math.OA

Remark on Serre $C^*$-algebras

We study non-commutative algebraic geometry of Artin, Serre and Tate in terms of the operator algebras. Namely, we define the Serre $C^*$-algebra $\mathcal{A}_X$ of a projective variety $X$ as the norm-closure of a representation of the twisted homogeneous coordinate ring of $X$ by the linear operators on a Hilbert space $\mathcal{H}$. It is proved that $X$ is homeomorphic to the space of all irreducible representations of the crossed product of $\mathcal{A}_X$ by an automorphism of $\mathcal{A}_X$. The case of rational elliptic curves $X$ is considered in detail.

2022-05-03math.AGmath.OA

有限算子代数相关的非交换亚纯函数的斜域

Skew fields of non-commutative meromorphic functions affiliated to finite operator algebras

James Madison University（詹姆斯麦迪逊大学）；University of Manitoba（曼尼托巴大学）；Ben-Gurion University of the Negev（内盖夫本古里安大学）

该研究将非交换亚纯函数斜域嵌入有限冯·诺依曼代数相关的闭算子*代数，通过构造自伴算子三元组证明Connes-Shlyakhtenko定义的Δ是Voiculescu非微态自由熵维数的严格上界，二者并不相等。

2026-10-16math.OA