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The generator rank of C⁎-algebras
Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2021-02-01 , DOI: 10.1016/j.jfa.2020.108874
Hannes Thiel

The invariant that assigns to a C*-algebra its minimal number of generators lacks natural permanence properties. In particular, it may increase when passing to ideals or inductive limits. It is therefore hard to compute this invariant directly. To obtain a better behaved theory, we not only ask if k generators exist, but also if such tuples are dense. This defines the generator rank, which we show has many of the permanence properties that are also satisfied by other noncommutative dimension theories. In particular, it does not increase when passing to ideals, quotients or inductive limits. The definition of the generator rank is analogous to that of the real rank, and we show that the latter always dominates the generator rank. The most interesting value of the generator rank is one, which means exactly that the generators form a generic set, that is, a dense G_delta-subset. We compute the generator rank of homgeneous C*-algebras, which allows us to deduce that certain AH-algebras have generator rank one. For example, every AF-algebra has generator rank one and therefore contains a dense set of generators.

中文翻译:

C⁎-代数的生成器秩

赋予 C*-代数最少生成器数量的不变量缺乏自然的永久性属性。特别是,当传递到理想或归纳极限时,它可能会增加。因此很难直接计算这个不变量。为了获得更好的理论,我们不仅要问是否存在 k 个生成器,还要问这样的元组是否密集。这定义了生成器秩,我们展示了它具有许多其他非交换维数理论也满足的永久性属性。特别是,当传递到理想、商或归纳极限时,它不会增加。生成器秩的定义类似于真实秩的定义,我们证明后者总是支配生成器秩。生成器秩最有趣的值是 1,这意味着生成器形成一个泛型集,也就是说,一个密集的 G_delta 子集。我们计算齐次 C*-代数的生成器秩,这允许我们推断某些 AH-代数的生成器秩为 1。例如,每个 AF 代数的生成器等级为 1,因此包含一组密集的生成器。
更新日期:2021-02-01
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