Flows and their symmetries on operator algebras

我们一直试图通过将算子代数上的流看作量子统计力学中的一个模型来发现和解决与流有关的问题。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个过程中获得的三个主要结果。罗林流：这是我们在统计力学中遇到的最远的一种流，但在数学上可能是最简单的一种，因为1-上循环几乎是同界的。尽管有许多已知的流具有这一性质的例子，我们已经证明了对于所有的Kirchberg代数，它们可能是唯一的，直到具有部分结果的上循环共轭(与O.Bratteli和D.W.Robinson)。证明源于对2×2矩阵的无限张量积上的移位自同构的另一个1-上循环性质及其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来归纳定义的流，并且被认为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)，但我们证明了，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。证明使用了来自统计力学的物理性质。流，对不变遗传子代数的限制，以及乘子的扰动：这一点在C^*-情况下没有被触及，尽管在一些W^*-情况下得到了众所周知的结果。这就归结为稳定C^*-代数上的流的余循环乘子问题。一个主要的困难在于这样一个事实，即这种协循环不是范数连续的。采用泛函分析的方法，证明了余循环可以在弱意义下用范数连续的余循环来逼近，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.