Lp Estimates in Non-commutative Probability and Analysis
非交换概率和分析中的 Lp 估计
基本信息
- 批准号:0301116
- 负责人:
- 金额:$ 12.52万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2003
- 资助国家:美国
- 起止时间:2003-05-15 至 2006-04-30
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
AbstractJungeThe starting point of this project on 'Lp estimates in noncommutative probability and analysis' are recent results and techniques from noncommutative martingales inequalities obtained by Pisier/Xu, Randrianantoanina, Junge and Junge/Xu. This new insight enables us to show a noncommutative analog the maximal ergodic theorem for completely positive maps and study square function inequalities in this setting (joint work with LeMerdy/Xu). The formulation and properties are motivated by the theory of Operator Spaces. On the other hand martingale inequalities are crucial in understanding independent, indiscernable and exchangeable sequences in noncommutative Lp spaces and properties of almost uniform convergence. Martingale inequalities and noncommutative probability are also fundamental tools in analyzing the operator space OH and its realization in the predual of type III von Neumann algebras. These techniques are similar to those used by Pisier/Shlyaktenko in proving the noncommutative version of Grothendieck's inequality. Surprising the analog of the 'little Grothendieck inequality' in the context of operator spaces only holds up to a logarithmic factor.Quantum mechanics and Heisenberg's uncertainty principle and mathematical models realizing these phenomena changed not only our perception of the world but also the mathematical discipline. Many noncommutative (=quantum) generalizations of classical mathematical theories for example the theory of quantum groups and noncommutative (=quantum) probability theory. Very interesting new phenomena and difficulties arise when adopting classical concepts to this noncommutative framework. Noncommutative measure theory and the theory of von Neumann algebras provide plenty of examples of genuinely new phenoma. Indeed, von Neumann's motivation for his work on operator algebras (now called von Neumann algebras) was to provide a good mathematical foundation for quantum mechanics. In this tradition the theory of Operator Spaces provides the right language for quantizing Banach spaces, a notion developed to describe the spaces of solutions of differential equations. For example, using this language it is now possible to talk about the expected exit time for a noncommutative domain although we can never see the 'points' of this domain. As a long term perspective these mathematical theories provide new features which may be used to understand phenomena in physics and other natural sciences.
本文的出发点是Pisier/Xu,Randrianantoanina,Junge和Junge/Xu从非交换鞅不等式中得到的最新结果和技巧。 这一新的见解使我们能够显示一个非交换模拟的最大遍历定理完全正的地图和研究平方函数不等式在这种设置(联合工作与LeMerdy/徐)。 公式和性质的动机是由算子空间理论。 另一方面,鞅不等式对于理解非交换Lp空间中独立的、不可分辨的和可交换的序列以及几乎一致收敛的性质至关重要。鞅不等式和非交换概率也是分析算子空间OH及其在III型冯诺依曼代数的预对偶中实现的基本工具。 这些技术类似于皮西尔/什利亚坚科在证明格罗滕迪克不等式的非交换版本中所使用的技术。 令人惊讶的是,在算子空间中,“小格罗滕迪克不等式”的类比只适用于对数因子。量子力学和海森堡的测不准原理以及实现这些现象的数学模型不仅改变了我们对世界的看法,也改变了数学学科。经典数学理论的许多非对易(=量子)推广,例如量子群理论和非对易(=量子)概率论。当采用经典概念到这个非对易框架时,会出现非常有趣的新现象和困难。非交换测度理论和冯·诺依曼代数理论为真正的新现象提供了大量的例子。事实上,冯诺依曼的动机,他的工作算子代数(现在称为冯诺依曼代数)是提供一个良好的数学基础,量子力学。 在这个传统中,算子空间理论为量化Banach空间提供了正确的语言,这是一个用来描述微分方程解的空间的概念。例如,使用这种语言,现在可以讨论一个非交换整环的期望退出时间,尽管我们永远看不到这个整环的“点”。作为一个长期的观点,这些数学理论提供了新的功能,可用于理解物理学和其他自然科学的现象。
项目成果
期刊论文数量(0)
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Marius Junge其他文献
Embeddings of symmetric operator spaces into Lp-spaces, 1 ≤ p < 2, on finite von Neumann algebras
- DOI:
10.1007/s11856-025-2743-0 - 发表时间:
2025-03-27 - 期刊:
- 影响因子:0.800
- 作者:
Jinghao Huang;Marius Junge;Fedor Sukochev;Dmitriy Zanin - 通讯作者:
Dmitriy Zanin
Some estimates on entropy numbers
- DOI:
10.1007/bf02760951 - 发表时间:
1993-10-01 - 期刊:
- 影响因子:0.800
- 作者:
Marius Junge;Martin Defant - 通讯作者:
Martin Defant
On the relation between completely bounded and (1,emcb/em)-summing maps with applications to quantum XOR games
关于完全有界映射与(1,emcb/em)-求和映射之间的关系及其在量子异或游戏中的应用
- DOI:
10.1016/j.jfa.2022.109708 - 发表时间:
2022-12-15 - 期刊:
- 影响因子:1.600
- 作者:
Marius Junge;Aleksander M. Kubicki;Carlos Palazuelos;Ignacio Villanueva - 通讯作者:
Ignacio Villanueva
Random variables in weak typep spaces
- DOI:
10.1007/bf01189933 - 发表时间:
1992-04-01 - 期刊:
- 影响因子:0.500
- 作者:
Martin Defant;Marius Junge - 通讯作者:
Marius Junge
On ?ℒ∞ structures of nuclear C * -algebras
- DOI:
10.1007/s00208-002-0384-7 - 发表时间:
2003-03-01 - 期刊:
- 影响因子:1.400
- 作者:
Marius Junge;Narutaka Ozawa;Zhong-Jin Ruan - 通讯作者:
Zhong-Jin Ruan
Marius Junge的其他文献
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{{ truncateString('Marius Junge', 18)}}的其他基金
CQIS: Operator algebra and Quantum Information Theory
CQIS:算子代数和量子信息论
- 批准号:
2247114 - 财政年份:2023
- 资助金额:
$ 12.52万 - 项目类别:
Standard Grant
Operator Algebra Theory in Applications
算子代数理论的应用
- 批准号:
1800872 - 财政年份:2018
- 资助金额:
$ 12.52万 - 项目类别:
Continuing Grant
Great Plains Operator Theory Symposium (GPOTS) 2016
大平原算子理论研讨会 (GPOTS) 2016
- 批准号:
1566648 - 财政年份:2016
- 资助金额:
$ 12.52万 - 项目类别:
Standard Grant
Operator algebras between theory and application
理论与应用之间的算子代数
- 批准号:
1501103 - 财政年份:2015
- 资助金额:
$ 12.52万 - 项目类别:
Continuing Grant
Applications of operator algebra theory to certain problems in analysis
算子代数理论在某些分析问题中的应用
- 批准号:
0901457 - 财政年份:2009
- 资助金额:
$ 12.52万 - 项目类别:
Continuing Grant
Noncommutative Hardy Spaces and Littlewood-Paley Theory
非交换 Hardy 空间和 Littlewood-Paley 理论
- 批准号:
0901009 - 财政年份:2009
- 资助金额:
$ 12.52万 - 项目类别:
Standard Grant
Quantum Probabilistic Methods in Operator Spaces and Applications
算子空间中的量子概率方法及其应用
- 批准号:
0556120 - 财政年份:2006
- 资助金额:
$ 12.52万 - 项目类别:
Standard Grant
Non-commutative Lp-spaces and their Connection to Probability and Operator Spaces
非交换 Lp 空间及其与概率和算子空间的联系
- 批准号:
0088928 - 财政年份:2000
- 资助金额:
$ 12.52万 - 项目类别:
Standard Grant
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