Flows and their symmetries on operator algebras

我们一直试图把算子代数作为量子统计力学的一个模型，来发现和解决算子代数上的流问题。我们在标题中提到对称性，因为我们预期这项研究可能会对相变做出贡献。虽然我们所希望的没有实现，我们将列出并解释在这个过程中获得的三个主要结果：Rohlin流：这是我们在统计力学中遇到的最远的一类流，但可能是数学上最简单的一类，因为1-上循环几乎是共边界的。虽然有许多已知的流的例子具有这一性质，我们已经证明，有所有的基希贝格代数，并指出，他们可能是唯一的上循环共轭的部分结果（O。Bratteli和D.W.罗宾逊）。这个证明起源于对2 × 2矩阵的无限张量积上的移位自同构的另一个1-上循环性质的研究，以及它对所谓规范不变部分的限制。AF流：这是一种可以基于矩阵代数上的流归纳定义的流，应该对应于经典模型而不是量子模型。一个基本的问题仍然没有解决，量子流是否真的远离经典流（或AF流），但我们表明，如果流看起来接近AF流的外观，那么它是接近的原则。证明使用了统计力学中的物理性质：流、对不变遗传子代数的限制和乘子的扰动：尽管在W^*-情形中有一些著名的结果，但在C^*-情形中没有涉及到这些。这归结为稳定C^*-代数上流的上循环乘子问题。一个主要的困难在于这样的上循环不是范数连续的。采用泛函分析的方法，证明了该上圈可以用弱范数连续上圈来逼近，从而达到了我们的目的。

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.