We study simply connected Lie groups $G$ for which the hull-kernel topology of the primitive ideal space $\mathrm{Prim}(G)$ of the group $C^{\ast }$-algebra $C^{\ast }(G)$ is $T_1$, that is, the finite subsets of $\mathrm{Prim}(G)$ are closed. Thus, we prove that $C^{\ast }(G)$ is AF-embeddable. To this end, we show that if $G$ is solvable and its action on the centre of $[G,G]$ has at least one imaginary weight, then $\mathrm{Prim}(G)$ has no nonempty quasi-compact open subsets. We prove in addition that connected locally compact groups with $T_1$ ideal spaces are strongly quasi-diagonal.

中文翻译：

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具有 T1 原始理想空间的李群的 AF 可嵌入性

我们研究单连通李群 $G$ 原始理想空间的壳核拓扑 $\mathrm{普里姆}(G)$ 组的 $C^{＊}$-代数 $C^{＊}(G)$ 是 ${吨}_1$，即有限子集 $\mathrm{普里姆}(G)$关闭。因此，我们证明$C^{＊}(G)$可嵌入 AF。为此，我们证明如果$G$ 是可解的，其作用在 $[G,G]$ 至少有一个虚权重，那么 $\mathrm{普里姆}(G)$没有非空的拟紧开子集。我们另外证明连接局部紧群与${吨}_1$ 理想空间是强拟对角空间。