算子空间中的量子概率方法

The interplay between operator spaces and quantum probability has started to emerge in the last years. Noncommutative Lp-spaces are at the intersection of these topics and play a crucial role in recent mathematical research motivated by concepts and problems from quantum physics. Even though it is not new, the noncommutative integration has recently regained considerable interest among researchers from different fields such as functional analysis, mathematical physics, quantum probability and quantum information. These "quantum mathematics" are at the frontier between theoretical physics and mathematics. The first interactions between them are already impressive, one can certainly expect that their full exploration will open a new vast avenue of perspectives and impact the future development of these areas..Some key results in the last ten years were obtained through sporadic explorations of interactions between these quantised theories. It would be appropriate to fully investigate and develop these links, in particular through the applications of noncommutative Lp-spaces, noncommutative martingale and matricial inequalities. These inequalities of quantum probabilistic nature have natural applications to theoretical and mathematical physics. The originality of this project lies in the systematic exploration of these interactions to build up a coherent program that goes far beyond the current state of the art. We will exploit particularly our original and novel approaches to these fields where noncommutative Lp-spaces have been rarely or have not been explored so far. We firmly believe that the research in these domains is at a crossroad, and that this is the right time to investigate them through the insight of noncommutative Lp-spaces. This project will initiate new studies across a wide spectrum of fields at the frontiers in noncommutative analysis..The first part of the project is to study operator spaces by quantum probabilistic methods. It deals with Grothendieck's program which requires notably the concrete realisation of operator spaces. Another aspect concerns the study of fundamental examples of completely bounded maps such as Fourier or Schur multipliers in view of applications. The second part goes conversely and treats the exploration of quantum probability by operator spaces. The most ambitious objective here is the creation of an analytic quantum stochastic integral which now seems possible thanks to the noncommutative martingale inequalities. This analytic theory of quantum stochastic integration should open a large field of applications exactly like It?'s integral in the classical probability theory. Markov dilations and harmonic analysis on semigroups will be also part of our objectives. Finally, exploiting the full potential of this interplay should allow us to solve fundamental open problems in these areas.

算子空间和量子概率的相互联系是近年来才开始的研究方向，这些领域源于量子物理。而非交换结构在近期也重新引起了很多不同学科领域研究者的兴趣。现在这些"量子化数学"是数学和理论物理两大学科的前沿性课题。我们将通过非交换Lp空间理论来全面研究和发展这种联系。项目的第一部分是用量子概率方法研究算子空间。这一方向包括一个Grothendieck研究计划；还有关于Fourier乘子完全有界性的研究及其应用。项目的第二部分是用算子空间方法来处理量子概率。这一方向最重要的目标是建立一个分析量子It? 积分,这将开辟一片广阔的应用前景，正如经典的It?积分一样；Markov 扩张和量子半群上的调和分析也是这一方向的研究目标。量子概率和算子空间的相互作用的开发将有助于我们解决这些领域中的一些重要开放性问题。本项目的新颖之处在于系统地发展并利用这种相互关系，从而建立起远远超过现有研究框架的持续性研究体系。

算子空间与量子概率的交叉研究近年来十分活跃，这些领域源于量子物理，其非交换结构近期重新引起了很多不同学科领域研究者的兴趣，它们正被有效地应用于其他领域如量子信息。本项目的研究内容为：1）用量子概率的方法研究算子空间的问题。将通过算子空间的具体实例研究Grothendieck纲领；研究完全有界Fourier-Schur乘子及其在逼近性质中的应用以及算子空间几何理论。2）用算子空间的理论和方法研究量子概率与非交换调和分析，主要目标是用非交换鞅不等式理论的最新成果建立量子随机积分的Lp理论，它将为量子概率开辟广阔的应用领域；研究量子Markov半群上的调和分析，包括经典和向量值的调和和分析理论。已取得的理论结果包括建立非交换Lp空间和非交换鞅不等式，系统研究量子环面上的调和分析理论，发展非交换调和分析以及算子半群上向量值的调和分析理论。这些结果都发表在国际知名的数学期刊上。我们坚信量子概率和算子空间的交叉研究正处在一个关键的发展时期，非交换Lp空间理论在这些领域的应用正当其时，本项目成果必将促进非交换分析研究的发展。