Quantization

自从量子力学出现以来，我们越来越多地研究非交换对象，但它们的内容范围很广。我们研究了各种函数空间，这些空间上的一些算子，哈密顿算子和狄拉克算子，算子间的关系和不等式，算子的代数，这些代数的自同构，自同构的群，群的作用和协同，量子群的表示等。主要研究了C^<**>-代数及其自同构内容中遍历理论中所谓Rohlin性质的量子化问题。由于我们必须使用范数拓扑，罗林性质的当前定义对代数和自同构施加了很强的限制（与冯·诺伊曼代数的情况相比），这可能表明进一步的推广是可取的。但只要我们坚持目前的定义，我们似乎已经得出了一般的结果。进一步的结果如下：对纯不变态丰度的结果进行了推广，这与经典力学和量子力学形成了鲜明的对比（Kishimoto）。研究了Hardy和Bergman空间等各种函数空间，Toeplitz算子等算子以及不变子空间的问题（Nakazi）。研究了有界解析函数的点分离问题。分析了各种模型的哈密顿量、正则交换关系的表示和量子群的表示（Arai）。尝试了von Neumann代数上有限群的协作用分类（Yamanouchi）。

Ever since the advent of quantum mechanics we have treated non-commutative objects more and more often but their contents range wildly. We investigate various function spaces, some operators on these spaces, Hamiltonians and Dirac operators, relations and inequalities among operators, algebras of operators, automorphisms of these algebras, groups of automorphisms, actions and coactions of groups, representations of quantum groups etc. The head investigator considered the problem of quantizing the so-called Rohlin property in ergodic theory in the contents of C^<**>-algebras and their automorphisms. Since we must use norm topology, the present definition of the Rohlin property imposes strong restrictions on the algebras and automorphisms (compared with the von Neumann algebra case), which may suggest a further generalization is desirable. But as far as we stick to the present definition, we seem to have reached general results.Further results are as follows : A generalization is attempted for a result of abundance of pure in-variant states which makes a sharp contrast between classical and quantum mechanics (Kishimoto). Various function spaces such as Hardy and Bergman spaces, such operators as Toeplitz operators and the problem of invariant subspaces are investigated (Nakazi). The problem of separation of points are studied for bounded analytic functions (Hayashi). Hamiltonians of various models, repre-sentations of canonical commutation relations, and representations of quantum groups are analized (Arai). Classification of coactions of finite groups on von Neumann algebras are attempted (Yamanouchi).