论文标题
Tracial状态和$ \ Mathbb {G} $ - 离散量子组的不变状态
Tracial States and $\mathbb{G}$-Invariant States of Discrete Quantum Groups
论文作者
论文摘要
我们研究了奇特状态和$ \ mathbb {g} $ - 在简化的$ c^*$ - 代数$ c_r（\ wideHat {\ mathbb {g}}）$上的不变状态。在这里，我们用$ \ wideHat {\ mathbb {g}} $表示其双重紧凑量量子组。我们的主要结果是，当$ c_r（\ wideHat {\ mathbb {g}}}）上的状态是tracial的，并且仅当它是$ \ mathbb {g} $ - 不变。这概括了一个非模块化离散量子组的已知事实，并建立在Kalantar，Kasprzak，Skalski和Vergnioux的工作基础上。因此，我们发现$ c_r（\ wideHat {\ mathbb {g}}）$是核的，并且仅当$ \ mathbb {g} $是可观的，并且仅当时才能接受奇特状态。由于C.-K。在离散案例中，NG和Viselter和Crann。另一个结果，我们证明了$ C_R（\ wideHat {\ Mathbb {g}}）$上的Tracial在$ \ wideHat {\ Mathbb {g}} _ f $上，其中$ \ Mathbb {g Mathbb {g} _f $是FurSternberg的cokernelg youncor。此外，考虑到某些假设，我们表征了$ c_r（\ wideHat {\ mathbb {g}}}）$在$ \ wideHat {\ wideHat {\ mathbb {g}} _ f $ is kac类型方面的存在。我们还根据（IDEMPOTENT）跟踪的唯一性，说明不是$ \ wideHat {\ mathbb {g}} _ f $等于$ \ wideHat {\ mathbb {g}} $的规范kac商。这些结果取决于以下结果，我们提供了证明：Sołtan的规范KAC商结构，无论是应用于通用还是减少的CQG $ C^*$ - $ \ wideHat {\ Mathbb {G}} $的代数（当后者允许一条痕迹），都会产生封闭的量子kac型Subgroum， $ \ wideHat {\ mathbb {g}} $。
We investigate the tracial states and $\mathbb{G}$-invariant states on the reduced $C^*$-algebra $C_r(\widehat{\mathbb{G}})$ of a discrete quantum group $\mathbb{G}$. Here, we denote its dual compact quantum group by $\widehat{\mathbb{G}}$. Our main result is that a state on $C_r(\widehat{\mathbb{G}})$ is tracial if and only if it is $\mathbb{G}$-invariant. This generalizes a known fact for unimodular discrete quantum groups and builds upon the work of Kalantar, Kasprzak, Skalski, and Vergnioux. As one consequence of this, we find that $C_r(\widehat{\mathbb{G}})$ is nuclear and admits a tracial state if and only if $\mathbb{G}$ is amenable. This resolves an open problem due to C.-K. Ng and Viselter, and Crann, in the discrete case. As another consequence, we prove that tracial states on $C_r(\widehat{\mathbb{G}})$ "concentrate" on $\widehat{\mathbb{G}}_F$, where $\mathbb{G}_F$ is the cokernel of the Furstenberg boundary. Furthermore, given certain assumptions, we characterize the existence of traces on $C_r(\widehat{\mathbb{G}})$ in terms of whether or not $\widehat{\mathbb{G}}_F$ is Kac type. We also characterize the uniqueness of (idempotent) traces in terms of whether not $\widehat{\mathbb{G}}_F$ is equal to the canonical Kac quotient of $\widehat{\mathbb{G}}$. These results rely on the following, of which we give proofs: Sołtan's canonical Kac quotient construction, whether it is applied to the universal or the reduced CQG $C^*$-algebra of $\widehat{\mathbb{G}}$ (when the latter admits a trace), yields the maximal Kac type closed quantum subgroup of $\widehat{\mathbb{G}}$.
