New horizons in operator algebras: finite-dimensional approximations and quantized function theory

算子代数的新视野：有限维近似和量化函数理论

• 批准号: RGPIN-2022-03600
• 负责人: Clouatre, Raphael
• 金额: $ 1.97万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758497
• 关键词: New; horizons; operator; algebras; finite

The proposed research program aims to discover new structure in the theory of operator algebras. Classically, these algebras came into being as a mathematical counterpart to the theory of quantum physics. The link between operator algebras and natural sciences continues to grow today, as witnessed in quantum information theory or quantum field theory for instance. Simply put, the connection between operator algebras and quantum physics is realized by symmetric operators between Euclidean spaces. Much of the sought-after structure of these operators can be unlocked upon considering the operator algebras (or C*-algebras) that they generate. What sets this research proposal apart from such classical investigations is that it is concerned with algebras of operators that lack the usual symmetry of quantum mechanical observables. In some applications the relevant operators happen to be a perturbation of a usual observable, so the resulting operator algebras are not susceptible to the tools designed to study C*-algebras. To elucidate the structure of these non self-adjoint operator algebras, I will employ innovative techniques in quantized function theory to implement finite-dimensional approximations. If we consider matrices to be the most basic objects, we may subsequently try to use them as building blocks for more complex algebras. This is the basic idea of a finite-dimensional approximation, a paradigm that has led to recent spectacular developments in the classification theory of C*-algebras. Some of the crucial insight involved in these advances was brought to light via a powerful functional analogy: one can view non-commutative algebras as consisting of functions, and exploit the resulting intuition coming from classical topology or dynamical systems theory. In spite of the unquestionable success of such a "quantized" theory of functions for C*-algebras, the corresponding non self-adjoint version has received measurably less attention. It is the purpose of this proposal to fill this gap. I will develop new tools in quantized function theory, a vibrant and rapidly evolving field with applications to systems and control theory, free probability and real algebraic geometry. The passage from C*-algebras to non self-adjoint ones should be mirrored by the set of non-commutative continuous functions collapsing to those that are in fact holomorphic. Classical intuition then suggests that this should result in a significant loss of flexibility and in a wealth of new rigidity phenomena. Towards achieving the above objectives, I have formulated an extensive scaffolding of concrete steps. The resulting numerous sub-problems of varied difficulties form the basis of my training plan for a diverse group of highly qualified personnel that I will recruit at all levels. Collaborations and interactions within my research group will be encouraged, and will foster an inclusive training environment.

拟议的研究计划旨在发现操作员代数理论中的新结构。从经典上讲，这些代数与量子物理学理论成为数学对应物。如量子信息理论或量子场理论所见证的那样，算法代数与自然科学之间的联系继续发展。简而言之，欧几里得空间之间的对称操作员实现了操作员代数和量子物理学之间的连接。考虑到它们产生的操作员代数（或C*-ergebras），可以解锁这些操作员的许多追求结构。这项研究建议与此类古典研究不同的是，它与缺乏量子机械可观察物的通常对称性的操作员代数有关。 在某些应用中，相关操作员恰好是通常可观察到的扰动，因此所得的操作员代数不容易受到旨在研究C*-Algebras的工具。为了阐明这些非自我伴侣操作员代数的结构，我将在量化功能理论中采用创新技术来实施有限的维近似值。如果我们将矩阵视为最基本的对象，则随后可能会尝试将其用作更复杂的代数的构建块。这是有限维近似的基本思想，这是一种范式，导致了C*-Algebras分类理论的最新发展。这些进步所涉及的一些关键见解是通过强大的功能类比来揭示的：一个人可以将非共同的代数视为由功能组成，并利用来自古典拓扑或动力学系统理论产生的直觉。尽管对于C* - 代数的“量化”功能理论毫无疑问，相应的非自我接合版本的关注量大大降低了。填补这一空白是该提议的目的。我将开发量化功能理论的新工具，该工具是一个充满活力且快速发展的领域，并应用于系统和控制理论，自由概率和实际代数几何形状。从c* - 代数到非自我伴侣的段落应通过一组非交通性连续函数倒塌的段落，这些功能崩溃了，实际上是霍明态的。然后，经典的直觉表明，这应该导致柔韧性大幅丧失和大量新的刚性现象。为了实现上述目标，我已经制定了大量混凝土步骤的脚手架。由此产生的各种困难的众多子问题构成了我为各种高素质人员组成的各种培训计划的基础，我将在各个层面上招募。将鼓励我的研究小组中的协作和互动，并将培养包容性的培训环境。