We study the dual relationship between quantum group convolution maps $L^1(\mathbb{G})\rightarrow L^{\infty}(\mathbb{G})$ and completely bounded multipliers of $\widehat{\mathbb{G}}$. For a large class of locally compact quantum groups $\mathbb{G}$ we completely isomorphically identify the mapping ideal of row Hilbert space factorizable convolution maps with $M_{cb}(L^1(\widehat{\mathbb{G}}))$, yielding a quantum Gilbert representation for completely bounded multipliers. We also identify the mapping ideals of completely integral and completely nuclear convolution maps, the latter case coinciding with $\ell^1(\widehat{b\mathbb{G}})$, where $b\mathbb{G}$ is the quantum Bohr compactification of $\mathbb{G}$. For quantum groups whose dual has bounded degree, we show that the completely compact convolution maps coincide with $C(b\mathbb{G})$. Our techniques comprise a mixture of operator space theory and abstract harmonic analysis, including Fubini tensor products, the non-commutative Grothendieck inequality, quantum Eberlein compactifications, and a suitable notion of quasi-SIN quantum group, which we introduce and exhibit examples from the bicrossed product construction. Our main results are new even in the setting of group von Neumann algebras $VN(G)$ for quasi-SIN locally compact groups $G$.

中文翻译：

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量子群乘子的映射理想

我们研究了量子群卷积映射 $L^1(\mathbb{G})\rightarrow L^{\infty}(\mathbb{G})$ 和 $\widehat{\mathbb{G} 的完全有界乘数之间的对偶关系}}$。对于一大类局部紧致量子群 $\mathbb{G}$ 我们完全同构地确定行 Hilbert 空间可分解卷积映射的映射理想与 $M_{cb}(L^1(\widehat{\mathbb{G}} ))$，产生完全有界乘数的量子吉尔伯特表示。我们还确定了完全积分和完全核卷积映射的映射理想，后一种情况与 $\ell^1(\widehat{b\mathbb{G}})$ 重合，其中 $b\mathbb{G}$ 是$\mathbb{G}$ 的量子玻尔紧缩。对于对偶有界度的量子群，我们证明了完全紧凑的卷积映射与 $C(b\mathbb{G})$ 重合。我们的技术包括算子空间理论和抽象调和分析的混合体，包括 Fubini 张量积、非交换格洛腾迪克不等式、量子 Eberlein 紧化和准 SIN 量子群的合适概念，我们介绍并展示了双交叉的例子产品建设。即使在准 SIN 局部紧致群 $G$ 的群冯诺依曼代数 $VN(G)$ 的设置中，我们的主要结果也是新的。