Complexity rank for C∗-algebras-猫眼课题宝

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2022

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作者：

A. Jaime;R. Willett

C-代数的复杂性秩是由第二作者和Yu引入的，用于UCT的应用：非常粗略地说，如果你可以重复地将C-代数切成两半最多n次，并且最终得到有限维的东西，那么这个秩最多为n。在本文中，我们研究了复杂性秩，也是一个弱的复杂性秩，我们引入，有弱复杂性秩最多可以被认为是“双色局部有限维”。我们首先证明了对于可分的、有单位元的和单的C-代数，弱复杂度秩1等价于核维数1和真实的秩0的合取。特别地，这表明所有核C-代数的UCT等价于弱复杂性秩与具有零K-理论群的基希贝格代数的复杂性秩相等。然而，我们也表明，使用K理论的障碍（扭转K1），弱复杂性秩1和复杂性秩1是不一样的。然后，我们使用Kirchberg-Phillips分类定理来计算所有UCT基希贝格代数的复杂度秩：它总是1或2，秩1的情况发生当且仅当K1-群是无挠的。夏威夷大学马诺阿分校，2565麦卡锡购物中心，凯勒401 A，檀香山，HI 96816，美国; ajaime@hawaii.edu。夏威夷大学马诺阿分校，2565麦卡锡购物中心，凯勒401 A，檀香山，HI 96816，美国; rufus@math.hawaii.edu。1 ar X iv：2 20 5. 04 70 4v 2 [ m at h. O A ] 1 1 O ct 2 02 2

Complexity rank for C∗-algebras was introduced by the second author and Yu for applications towards the UCT: very roughly, this rank is at most n if you can repeatedly cut the C∗-algebra in half at most n times, and end up with something finite dimensional. In this paper, we study complexity rank, and also a weak complexity rank that we introduce; having weak complexity rank at most one can be thought of as ‘two-colored local finite-dimensionality’. We first show that for separable, unital, and simple C∗-algebras, weak complexity rank one is equivalent to the conjunction of nuclear dimension one and real rank zero. In particular, this shows that the UCT for all nuclear C∗-algebras is equivalent to equality of the weak complexity rank and the complexity ranks for Kirchberg algebras with zero K-theory groups. However, we also show using a K-theoretic obstruction (torsion in K1) that weak complexity rank one and complexity rank one are not the same in general. We then use the Kirchberg-Phillips classification theorem to compute the complexity rank of all UCT Kirchberg algebras: it is always one or two, with the rank one case occurring if and only if the K1-group is torsion free. University of Hawai‘i at Mānoa, 2565 McCarthy Mall, Keller 401A, Honolulu, HI 96816, USA; ajaime@hawaii.edu. University of Hawai‘i at Mānoa, 2565 McCarthy Mall, Keller 401A, Honolulu, HI 96816, USA; rufus@math.hawaii.edu. 1 ar X iv :2 20 5. 04 70 4v 2 [ m at h. O A ] 1 1 O ct 2 02 2

Complexity rank for C∗-algebras

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