Fundamental Decomposition in Finite von Neumann Algebras

随着量子力学的出现，希尔伯特空间上算子的研究变得重要起来，但除此之外，对这些算子的理解已被证明对许多数学领域的进步至关重要。历史上，研究和理解这些算子的一种方法是基于谱分解将它们分解成更简单的分量。这包括描述运算符的某些部分，这些部分的行为类似于特定数字的乘法，并解释这些部分如何组合成整体。该项目的一个主要目标是促进对大型操作符的理解。另一个主要目标是研究在各种量子力学模型中出现的算子族，根据关于无限维物体的有限维近似的某些深刻的数学猜想。更具体地说，首席研究员及其合作者近年来在有限冯诺伊曼代数的非自伴随元素的谱分解结果方面取得了进展。这些以上三角形式为中心，类似于Issai Schur关于矩阵的经典结果，并且利用和扩展了Haagerup和Schultz最近发现的关于超不变子空间的结果。特别建议的项目包括(a)研究有界算子的范数收敛性质和(b)将谱分布和上三角形式的结果推广到无界附属算子。在相关方向上，首席研究员将研究迹冯·诺伊曼代数元素的超不变子空间问题，并研究类似于海森堡关系的默里-冯·诺伊曼难题。拟议研究的第二个领域涉及无双性和无双产品的概念。提议研究的第三个领域涉及量子相关性。首席研究员最近的研究结果显示，五个输入和两个输出的量子相关集的非封闭性打开了对这些小案例的新理解的大门，这是首席研究员建议追求的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The study of operators on Hilbert space became important with the advent of Quantum Mechanics, but in addition, understanding of these operators has proven to be vital to progress in many areas of mathematics. Historically, a method of studying and understanding such operators is to break them down into simpler components, based on spectral decomposition. This consists of describing parts of the operator that behave like multiplication by certain numbers, and to explain how these parts assemble into the whole. One major goal of this project is to advance such understanding of large classes of operators. Another major goal is to study families of operators that arise in various quantum mechanical models, in light of certain deep mathematical conjectures regarding finite dimensional approximations of infinite dimensional objects.More specifically, the principal investigator, together with collaborators, has made advances in recent years on spectral decomposition results for non-selfadjoint elements of finite von Neumann algebras. These are centered around upper triangular forms, analogous to the classical results of Issai Schur for matrices, and both utilize and extend results about hyperinvariant subspaces found recently by Haagerup and Schultz. Particular proposed projects include (a) studying norm convergence properties of bounded operators and (b) extending spectral distribution and upper-triangular form results to unbounded affiliated operators. In related directions, the principal investigator will work on the hyperinvariant subspace problem for elements of tracial von Neumann algebras and to investigate the Murray--von Neumann puzzle, which is akin to the Heisenberg relations. A second area of proposed research concerns the notion of bi-freeness and bi-free products. A third area of proposed research concerns quantum correlations. Recent results of the principal investigator showing non-closedness of the set of quantum correlations for five inputs and two outputs open the door to new understanding of these small cases, which the principal investigator proposes to pursue.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.