$C^*$-algebras with finite complexity
Arturo Jaime, Rufus Willett

TL;DR
This paper explores the concept of complexity rank in $C^*$-algebras, introduces a weak variant, and characterizes their ranks in relation to properties like nuclear dimension, real rank, and $K$-theory, with implications for the UCT.
Contribution
It introduces the weak complexity rank, establishes its equivalence with known properties in certain cases, and computes the complexity rank for all UCT Kirchberg algebras.
Findings
Weak complexity rank one equals nuclear dimension one and real rank zero.
Complexity rank of UCT Kirchberg algebras is either one or two.
Weak complexity rank and complexity rank differ due to $K$-theoretic obstructions.
Abstract
Complexity rank for -algebras was introduced by the second author and Yu for applications towards the UCT: very roughly, this rank is at most if you can repeatedly cut the -algebra in half at most 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 -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 -algebras is equivalent to equality of the weak complexity rank and the complexity ranks for Kirchberg algebras with zero -theory groups. However, we also show using a -theoretic obstruction…
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topics in Algebra · Algebraic structures and combinatorial models
