Flows and their symmetries on operator algebras

算子代数上的流及其对称性

• 批准号: 16540181
• 负责人: KISHIMOTO Akitaka
• 金额: $ 2.54万
• 依托单位: Hokkaido University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16540181/
• 关键词: C^*-algebra; cocyle; Kirchberg algebra; Rohlin property; AF flow; multiplier; approximately inner; core symmetry; グラフ極限; コンヌスペクトル; クンツ環; K理論; ローリン流れ; C*環; AF環; 極大可換部分環; 内部近似性; C^*環; C^*-algebra; cocyle; Kirchberg algebra; Rohlin property; AF flow; multiplier; approximately inner; core symmetry; グラフ極限; コンヌスペクトル; クンツ環; K理論; ローリン流れ; C*環; AF環; 極大可換部分環; 内部近似性; C^*環

We have been trying to find and solve problems concerning flows on operator algebras by regarding them as a model in quantum statistical mechanics. We referred to symmetries in the title because we anticipated a contribution to phase transition might follow from this research. Although what we hoped for was not attained, we shall list and explain three major results obtained in this process.Rohlin flows: This is the farthest kind of flows from what we encounter in statistical mechanics but probably is the simplest kind mathematically because the 1-cocycles are almost coboundary. Although there are known many examples of flows with this property, we have shown that there are for all Kirchberg algebras and remarked that they may be unique up to cocycle conjugacy with a partial result (with O. Bratteli and D.W. Robinson). The proof originates in the research on another 1-cocycle property of the shift automorphism on the infinite tensor product of two-by-two matrices and its restriction to a so-called gauge-invariant part.AF flows: This is a flow which can be inductively defined based on flows on matrix algebras and is supposed to correspond to a classical model instead of a quantum model. A basic question is still unsolved of whether a quantum flow is really far from a classical flow (or AF flow), but we showed that if the flow looks close to an AF flow on appearance then it is close on principle. The proof uses physical properties derived from statistical mechanics.Flows, restrictions to invariant hereditary subalgebras, and perturbations by multipliers: This had not been touched upon in the C^*-case in spite of the well-known results in some the W^*-case. This reduces to problems on cocycle multipliers for a flow on a stable C^*-algebra. A main difficulty lies in the fact such a cocycle is not norm-continuous. By adopting functional analytic methods, we showed that the cocycle can be approximated by a norm-continuous cocycle in a weak sense, thus achieving our goal.

我们一直在尝试通过将它们作为量子统计力学的模型来查找和解决有关操作员代数流的问题。我们提到标题中的对称性，因为我们预计这项研究可能会对相过渡产生贡献。尽管我们希望获得的目的是未达到的，但我们将列出并解释在此过程中获得的三个主要结果。罗林流：这是与我们在统计力学中遇到的最远的流动，但在数学上可能是最简单的，因为1个循环的1几乎是同时的。尽管有许多与该特性流有关的示例，但我们已经证明了所有Kirchberg代数都有许多代数，并指出它们可能是独特的，可以与Cocycle cocycle cocycle结合（O. Bratteli和D.W. Robinson）。该证明起源于对自动形态的另一个1个循环特性的研究，这是对二次矩阵的无限张量产物及其限制到所谓的量规不变部分的限制。流动：这是一个流动：可以基于基于Matrix Algebras和Quantrix Algebras和Quantrix Algge and vental模型的流量来归纳。一个基本的问题仍未解决量子流是否远非经典流（或AF流），但我们表明，如果流动看起来接近外观上的AF流，则原理是接近的。该证明使用统计力学得出的物理特性。流量，不变的遗传亚呼骨的限制以及乘数的扰动：尽管有一些众所周知的w^** - 尽管有一些众所周知的情况，但在C^* - 尚未涉及。这减少了在稳定C^* - 代数上流动的Cocycle乘数上的问题。主要的困难在于这样的合子并不是规范性的。通过采用功能分析方法，我们表明，可以通过弱意义上的规范连续的共同体来近似合过程，从而实现了我们的目标。