关于顺从与余顺从的余理想
On Amenable and Coamenable Coideals
AI总结：
基于类群投影研究离散量子群右余理想的顺从性与余顺从性，通过引入不变态刻画相对顺从性，并建立余顺从紧拟子群与幂等态的一一对应，推进了余理想对偶性理论。
摘要：
我们研究离散量子群的右余理想 $\widetilde{N}_P\subseteq \ell^\infty(\mathbb{G})$ 的相对顺从性与顺从性，基于其类群投影 $P$ 进行刻画。我们建立了 $P$-左不变态的概念，并用它来刻画相对顺从性。我们还发展了紧拟子群 $N_\omega\subseteq L^\infty(\widehat{\mathbb{G}})$ 的余顺从性概念，推广了 Kalantar、Kasprzak、Skalski 和 Vergnioux 定义的商的余顺从性，其中 $\widehat{\mathbb{G}}$ 是 $\mathbb{G}$ 的紧对偶。特别地，我们证明了 $\widehat{\mathbb{G}}$ 的余顺从紧拟子群与既约 $C^*$-代数 $C_r(\widehat{\mathbb{G}})$ 上的幂等态一一对应。我们利用此工作获得了关于 $\ell^\infty(\mathbb{G})$ 中余理想的相对顺从性与顺从性，以及 $L^\infty(\widehat{\mathbb{G}})$ 中余对偶余理想的余顺从性之间对偶性的结果，推进了 Kalantar 等人的一个问题。
英文摘要：
We study relative amenability and amenability of a right coideal $\widetilde{N}_P\subseteq \ell^\infty(\mathbb{G})$ of a discrete quantum group in terms of its group-like projection $P$. We establish a notion of a $P$-left invariant state and use it to characterize relative amenability. We also develop a notion of coamenability of a compact quasi-subgroup $N_ω\subseteq L^\infty(\widehat{\mathbb{G}})$ that generalizes coamenability of a quotient as defined by Kalantar, Kasprzak, Skalski, and Vergnioux, where $\widehat{\mathbb{G}}$ is the compact dual of $\mathbb{G}$. In particular, we establish that the coamenable compact quasi-subgroups of $\widehat{\mathbb{G}}$ are in one-to-one correspondence with the idempotent states on the reduced $C^*$-algebra $C_r(\widehat{\mathbb{G}})$. We use this work to obtain results for the duality between relative amenability and amenability of coideals in $\ell^\infty(\mathbb{G})$ and coamenability of their codual coideals in $L^\infty(\widehat{\mathbb{G}})$, making progress towards a question of Kalantar et al{.}.
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