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而产生的。算符代数和自然科学之间的联系在今天继续增长，例如量子信息论或量子场论就证明了这一点。简单地说，算子代数和量子物理之间的联系是通过欧几里得空间之间的对称算子来实现的。只要考虑到它们生成的算子代数(或C*-代数)，这些算子的许多受欢迎的结构就可以被解锁。使这项研究计划有别于这类经典研究的是，它涉及的是缺乏量子力学可观察到的通常对称性的算子的代数。在某些应用中，相关算子恰好是通常可观测的算子的扰动，因此所得到的算子代数不受用于研究C*-代数的工具的影响。为了阐明这些非自伴算子代数的结构，我将使用量子化函数论中的创新技术来实现有限维近似。如果我们认为矩阵是最基本的对象，我们随后可能会尝试将它们用作更复杂的代数的构建块。这是有限维近似的基本思想，这一范例导致了C*-代数分类理论最近的惊人发展。这些进展中涉及的一些关键见解通过一个强大的泛函类比被揭示出来：人们可以将非交换代数视为由函数组成，并利用来自经典拓扑学或动力系统理论的直觉。尽管这种用于C*-代数的“量子化”函数理论取得了无可置疑的成功，但相应的非自伴形式却受到了相当少的关注。这项提议的目的是填补这一空白。我将开发量子化函数论的新工具，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、自由概率和实代数几何。从C~*-代数到非自共轭C~*-代数的过程应该被非交换连续函数集映射成实际上是全纯的函数集。然后，经典直觉表明，这应该会导致灵活性的显著丧失，并导致大量新的僵化现象。为了实现上述目标，我制定了广泛的具体步骤。由此产生的各种困难的许多子问题构成了我对各种高素质人员的培训计划的基础，我将在各级招聘这些人员。我的研究小组内部的合作和互动将得到鼓励，并将促进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.