论文标题
用$ \ operatatorName {\ mathsf {l}^{\！\ infty}}}（\ mathbb {g}）$一个因素的示例
Examples of compact quantum groups with $\operatorname{\mathsf{L}^{\!\infty}}(\mathbb{G})$ a factor
论文作者
论文摘要
对于每个$λ\ in \ left] 0,1 \右] $，我们展示了一个不可数的紧凑量量子组$ \ mathbb {g} $，以至于von neumann algebra $ \ mathsf {l}^{\！与可分离的预期。我们还表明，$ \ mathrm {iii} _0 $的许多注射因素作为$ \ mathsf {l}^{\！\ infty}（\ Mathbb {g}）$用于某些紧凑型量子组$ \ Mathbb {g} $。为了区分我们的示例，我们介绍了与von Neumann代数在Connes不变的$ t $上建立的比例组相关的不变性，并调查了$ \ Mathbb {g} $的不变性之间的联系，而Connes novariants $ t（\ Mathsf {\ mathsf {l}^^^\ f tyty^g}（\！ $ s（\ mathsf {l}^{\！\ infty}（\ mathbb {g}））$。在最后一部分中，我们显示了类型$ \ mathrm {i} $的因素，无法作为$ \ Mathsf {l}^{\！\！\ infty}（\ Mathbb {g}）$，对于非紧凑型量子组$ \ Mathbb {g} $。
For each $λ\in\left]0,1\right]$ we exhibit an uncountable family of compact quantum groups $\mathbb{G}$ such that the von Neumann algebra $\mathsf{L}^{\!\infty}(\mathbb{G})$ is the injective factor of type $\mathrm{III}_λ$ with separable predual. We also show that uncountably many injective factors of type $\mathrm{III}_0$ arise as $\mathsf{L}^{\!\infty}(\mathbb{G})$ for some compact quantum group $\mathbb{G}$. To distinguish between our examples we introduce invariants related to the scaling group modeled on the Connes invariant $T$ for von Neumann algebras and investigate the connection between so obtained invariants of $\mathbb{G}$ and the Connes invariants $T(\mathsf{L}^{\!\infty}(\mathbb{G}))$, $S(\mathsf{L}^{\!\infty}(\mathbb{G}))$. In the final section we show that factors of type $\mathrm{I}$ cannot be obtained as $\mathsf{L}^{\!\infty}(\mathbb{G})$ for a non-trivial compact quantum group $\mathbb{G}$.
