发表机构
Politecnico di Milano; Università di Roma Tor Vergata; University College Yonsei University; Hankyong National University（米兰理工大学; 罗马第二大学; 延世大学校; 韩京国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究C*-代数上态的支配性与等价性的非交换推广，推导相关Radon-Nikodym导数，探讨群作用下拟不变态的性质，给出群作用的酉实现并与Pedersen-Takesaki构造对比。
AI 中文摘要
我们研究测度论中态的支配性与等价性在C*-代数上的非交换推广，以探究群作用下的拟不变性。对于被支配的态，我们推导得到一个无界的“Radon-Nikodym”导数，它附属于支配态的GNS表示的交换子代数。值得注意的是，这种支配性通常不具备传递性，因为对应的闭算子的乘积可能无法闭化。当考虑*自同构的群作用时，固定拟不变态的轨道完全由相互等价的态构成，但轨道闭包可能包含奇异态，这意味着拟不变态的集合在凸组合下是闭的，但在拓扑意义上并非闭集。本文还给出了群作用在GNS希尔伯特空间上的酉实现，推广了协变表示，并将该方法与Pedersen-Takesaki构造进行了比较，后者的Radon-Nikodym导数位于中心化子而非交换子中。
英文摘要
We study the noncommutative generalization of measure-theoretic dominance and equivalence of states on $C^*$-algebras to explore quasi-invariance under group actions. For a dominated state, we derive an unbounded "Radon-Nikodym" derivative affiliated with the commutant algebra of the dominating state's GNS representation. Interestingly, this dominance is generally non-transitive because the product of the corresponding closed operators can be non-closable. When looking at group actions by $*$-automorphisms, the orbit of a fixed quasi-invariant state consists entirely of mutually equivalent states. However, the orbit closure may contain singular states, meaning the set of quasi-invariant states is closed under convex combinations but not topologically closed. The paper also provides a unitary implementation of the group action on the GNS Hilbert space-generalizing covariant representations and compares this approach with the Pedersen-Takesaki construction, where the Radon-Nikodym derivative sits in the centraliser instead of the commutant.
Comments35 pages
