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

提出的研究计划旨在发现算子代数理论中的新结构。经典地说，这些代数是作为量子物理理论的数学对应物而产生的。今天，算子代数与自然科学之间的联系仍在继续发展，例如量子信息论或量子场论就是例证。简而言之，算子代数与量子物理之间的联系是通过欧几里德空间之间的对称算子来实现的。只要考虑它们生成的算子代数（或C*-代数），就可以解开这些算子的许多受欢迎的结构。这个研究方案与这些经典研究的不同之处在于，它关注的是缺乏量子力学观测值通常对称性的算子代数。在某些应用中，相关的算符恰好是通常可观测值的扰动，因此所得的算符代数不容易受到设计用于研究C*-代数的工具的影响。为了阐明这些非自伴随算子代数的结构，我将采用量化函数理论中的创新技术来实现有限维逼近。如果我们认为矩阵是最基本的对象，我们随后可以尝试将它们用作更复杂代数的构建块。这是有限维近似的基本思想，这种范式导致了最近C*-代数分类理论的惊人发展。这些进步中涉及的一些重要见解是通过一个强大的泛函类比揭示出来的：人们可以将非交换代数视为由函数组成的代数，并利用来自经典拓扑或动力系统理论的直觉。尽管C*-代数的这种“量子化”函数理论取得了无可争议的成功，但相应的非自伴随版本却得到了相当少的关注。这一建议的目的就是填补这一空白。我将开发量化函数理论的新工具，这是一个充满活力和快速发展的领域，应用于系统和控制理论，自由概率和真正的代数几何。从C*-代数到非自伴随代数的过渡应该被非交换连续函数的集合所反映，这些函数实际上是全纯的。经典直觉认为，这将导致灵活性的显著丧失，并产生大量新的刚性现象。为了实现上述目标，我已制定了一系列具体步骤。由此产生的各种困难的许多子问题构成了我的培训计划的基础，我将在各个级别招聘各种各样的高素质人员。我的研究小组内的合作和互动将被鼓励，并将培养一个包容性的培训环境。

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.