AI 中文总结
该研究刻画了第二可数局部紧量子群的可和性，结合其对偶量子群的逼近性质，构造了受时间两端行为控制的卷积半群，相关结果在经典局部紧群情形下也属首次。
AI 中文摘要
我们针对第二可数局部紧量子群$\boldsymbol{\text{G}}$，建立了一种基于$\boldsymbol{\text{G}}$上态的卷积半群长时间行为的可和性简单刻画。随后，我们研究了更深入的问题：将$\boldsymbol{\text{G}}$的可和性与其对偶量子群$\boldsymbol{\text{\text{G}}}$的其他逼近性质相结合。具体而言，我们刻画了$\boldsymbol{\text{G}}$的（“强”）可和性与$\boldsymbol{\text{\text{G}}}$不具有性质（T）或具有哈盖鲁普性质的组合情况。这些结果即使在局部紧群的经典情形下似乎也是全新的，它们在$\boldsymbol{\text{G}}$上构造了卷积半群，其行为在时间趋近于零和时间趋近于无穷时均受到控制。
英文摘要
We establish a simple characterization of amenability of second countable locally compact quantum groups $\mathbb{G}$ in terms of the long-time behavior of convolution semigroups of states on $\mathbb{G}$. We then investigate the deeper problem of combining amenability of $\mathbb{G}$ with other approximation properties of the dual quantum group $\hat{\mathbb{G}}$. Specifically, we characterize the combination of ("strong") amenability of $\mathbb{G}$ with either the failure of property (T) or the presence of the Haagerup property for $\hat{\mathbb{G}}$. These results, which appear to be new even in the classical setting of locally compact groups, produce convolution semigroups on $\mathbb{G}$ whose behavior is controlled both as time goes to zero and as time goes to infinity.
Comments17 pages. Comments are welcome!
