AI 中文总结
研究保态幺正完全正映射空间拓扑，分析保态条件期望子空间拓扑，应用于子代数结构类，证明相关性质，还研究了特定态下非保态条件期望像的子代数空间，得出部分代数此类子代数构成开且稠密子集的结论。
AI 中文摘要
在本文中，我们确定了保态幺正完全正映射空间上的几种自然拓扑是一致的，并使其成为一个波兰空间。然后我们专注于保态条件期望的子空间，并详细分析其拓扑，恢复了哈格鲁普 - 温斯洛的结果，即它与冯·诺依曼子代数空间上的埃弗罗斯 - 马雷夏尔拓扑一致。这种对应关系随后被应用于子代数的结构类，包括顺从的、哈格鲁普的和弱顺从的子代数。我们证明了存在保态条件期望的顺从子代数的闭性，并分析了考林 - 哈格鲁普常数作为子代数函数时的半连续性和连续性的失效。最后，我们研究了对于固定的忠实正规态不是保态条件期望像的冯·诺依曼子代数空间。对于几类重要的冯·诺依曼代数，如\(0\leq\lambda\lt1\)的\({\rm III}_\lambda\)型因子和具有无限维中心化子态的\({\rm III}_1\)型因子，我们表明缺乏保态条件期望的子代数形成一个开且稠密的子集。因此，在这些情况下，一般的子代数不是保态条件期望的范围。
英文摘要
In this paper, we establish that several natural topologies on the space of state-preserving unital completely positive maps coincide and that make it a Polish space. We then focus on the subspace of state-preserving conditional expectations and analyse its topology in detail, recovering the Haagerup-Winslow result that it aligns with the Effros-Maréchal topology on the space of von Neumann subalgebras. This correspondence is then applied to structural classes of subalgebras, including amenable, Haagerup and weakly amenable subalgebras. Among other consequences, we demonstrate the closedness of amenable subalgebras admitting state-preserving conditional expectations and analyze the semicontinuity and failure of continuity of the Cowling-Haagerup constant as a function on subalgebras. Finally, we investigate the space of von Neumann subalgebras that are not the image of state preserving conditional expectations for a fixed faithful normal state. For several important classes of von Neumann algebras, such as type ${\rm III}_λ$ factors with $0 \le λ< 1$ and type ${\rm III}_1$ factors with a state whose centraliser is infinite dimensional, we show that the subalgebras lacking state-preserving conditional expectations form an open and dense subset. Thus, in these settings, the generic subalgebra is not the range of state preserving conditional expectation.
Comments23 pages
