Examples of compact quantum groups with L 8 ( G ) a factor
Examples of compact quantum groups with L 8 ( G ) a factor
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具有 L 8 ( G ) 因子的紧量子群示例
DOI：
10.1016/j.jfa.2023.110297
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发表时间：
2024
影响因子：
1.7
通讯作者：
Krajczok J
中科院分区：
文献类型：
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作者：
Krajczok J
对每个λ∈ [0，1]，我们证明了一个不可数紧量子群族G，使得von Neumann代数L∞（G）是具有可分预对偶的III型λ的内射因子。我们还证明了对于某个紧量子群G，有不可数多的III0型内射因子作为L∞（G）出现.我们引入自然不变量的量子群相关的标度群，仿照康纳斯不变量T的冯诺依曼代数。我们在示例中计算这些不变量的值，这使我们能够区分它们。我们还研究了它们与Connes不变量T（L∞（G）），S（L∞（G））的关系.在最后一节中，我们证明了对于紧致量子群G，vonNeumann代数L∞（G）不能有B（H）形式的直和项，其中dim H=∞.特别地，对于一个非平凡的紧致量子群G，I型因子（任何维数）不能作为L∞（G）得到。
Abstract For each λ∈] 0, 1] we exhibit an uncountable family of compact quantum groups G such that the von Neumann algebra L∞(G) is the injective factor of type III λ with separable predual. We also show that uncountably many injective factors of type III 0 arise as L∞(G) for some compact quantum group G. We introduce natural invariants of quantum groups related to the scaling group, modeled on the Connes invariant T for von Neumann algebras. We compute the values of these invariants in our examples, which allows us to distinguish between them. We also investigate their connection to the Connes invariants T (L∞(G)), S (L∞(G)). In the final section we show that for a compact quantum group G the von Neumann algebra L∞(G) cannot have a direct summand of the form B (H) with dim H=∞. In particular, factors of type I (of any dimension) cannot be obtained as L∞(G) for a non-trivial compact quantum group G.