Flows and their symmetries on operator algebras

我们一直试图通过将算子代数视为量子统计力学中的一个模型来寻找和解决算子代数上的流动问题。我们在标题中提到了对称性，因为我们预计这项研究可能会对相变做出贡献。虽然我们没有达到我们所希望的，但我们将列出并解释在这一过程中取得的三个主要成果。罗林流：这是我们在统计力学中遇到的最远的一种流，但可能是数学上最简单的一种，因为1环几乎是共边界的。虽然已知有许多具有这种性质的流的例子，但我们已经证明了所有Kirchberg代数都有这种性质，并指出它们可能是唯一的，直到有一个部分结果的循环共轭（与O. Bratteli和D.W. Robinson）。这个证明来源于对2乘2矩阵的无限张量积上的移位自同构的另一个1-环性质及其对所谓的规范不变部分的限制的研究。AF流：这是一种可以根据矩阵代数上的流归纳定义的流，并且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，即量子流是否真的远离经典流（或AF流），但我们表明，如果流在外观上看起来接近AF流，那么它在原理上是接近的。这个证明使用了统计力学的物理性质。流动、对不变遗传子代数的限制和乘子的扰动：这在C^*-情况中没有涉及，尽管在W^*-情况中有一些众所周知的结果。这就简化为稳定C^*-代数流的循环乘子问题。一个主要的困难在于这样的循环不是常态连续的。利用泛函解析的方法，我们证明了循环可以用一个弱意义上的范数连续循环来近似，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.