$E$-理论的Milnor正合列
A Milnor exact sequence for $E$-theory
AI总结：
该研究刻画了可分$C^*$-代数上$E$-理论群的幻影类，通过形状系统的导出逆极限建立了无需UCT与核性假设的Milnor正合列，并验证了其与已有结果的一致性。
AI中文摘要：
对于可分$C^*$-代数$A$和$B$，$E$-理论群$E(A,B)$具有一种自然的、通常非豪斯多夫的第二可数群拓扑，当且仅当零元的闭包$\boldsymbol{\overline{\{0\}}} \subseteq E(A,B)$平凡时，该拓扑是豪斯多夫的。$\overline{\{0\}}$中的非零元是幻影类，对所有取值于豪斯多夫空间的连续不变量都不可见。我们将$\overline{\{0\}}$刻画为$SA \otimes \mathcal{K}$的任意形状系统$(C_n)$的导出逆极限，从而对所有可分$A$和$B$得到自然的Milnor正合列：$0 \to {\varprojlim}^1 [C_n, S^2B \otimes \mathcal{K}] \to E(A,B) \to \varprojlim [C_n, SB \otimes \mathcal{K}] \to 0$，无需UCT或核性假设。当$A$是核代数时，该正合列与Willett-Yu受控-$KK$的Milnor正合列一致；当$A$满足UCT时（无论是否为核代数），其${\varprojlim}^1$项是纯扩张群$\mathrm{Pext}^1_{\mathbb{Z}}(K_{*+1}(A), K_*(B))$，与Schochet对Kasparov群的精细结构描述一致。
英文摘要：
For separable $C^*$-algebras $A$ and $B$, the $E$-theory group $E(A,B)$ carries a natural, generally non-Hausdorff, second-countable group topology, Hausdorff exactly when the closure of zero $\overline{\{0\}} \subseteq E(A,B)$ is trivial. The nonzero elements of $\overline{\{0\}}$ are the phantom classes, invisible to every continuous Hausdorff-valued invariant. We identify $\overline{\{0\}}$ as a derived inverse limit over any shape system $(C_n)$ of $SA \otimes \mathcal{K}$, giving a natural Milnor sequence $0 \to {\varprojlim}^1 [C_n, S^2B \otimes \mathcal{K}] \to E(A,B) \to \varprojlim [C_n, SB \otimes \mathcal{K}] \to 0$ for all separable $A$ and $B$, with no UCT or nuclearity hypothesis. For nuclear $A$ it agrees with the Willett-Yu controlled-$KK$ Milnor sequence, and when $A$ satisfies the UCT, nuclear or not, its ${\varprojlim}^1$ term is the pure-extension group $\mathrm{Pext}^1_{\mathbb{Z}}(K_{*+1}(A), K_*(B))$, as in Schochet's fine-structure description of the Kasparov groups.
