The radius of comparison for actions of Z^d on simple AH algebras

TL;DR

The paper addresses how the radius of comparison ${\rm rc}$ behaves under crossed products by countable discrete groups, particularly actions of $\mathbb{Z}^d$. It develops a constructive method to realize any pair $(r,r')$ with $0\le r'\le r\le \infty$ by producing a simple unital AH algebra $A$ with stable rank one and a pointwise outer action $\alpha$ such that ${\rm rc}(A)=r$ and ${\rm rc}(A\rtimes_{\alpha}\mathbb{Z}^d)=r'$. The main contribution is a diagram-merging construction that yields independent radii in $A$ and its crossed product, together with detailed computations of ${\rm rc}$ for both, demonstrating that no simple formula governs ${\rm rc}$ under crossed products. This work broadens understanding beyond Elliott-classifiable cases, showing that radius of comparison can decrease under crossed-product formation and that its behavior under discrete group actions is subtle and case-dependent.

Abstract

Given 0 \leq r' \leq r \leq \infty, and d \in N, we construct a simple unital AH algebra A with stable rank one, and a pointwise outer action α: Z^d \to Aut(A), such that rc(A)=r and rc (A \rtimes_α Z^d)=r'.

Theorems & Definitions (19)

• Theorem 1.1
• Remark 1.2
• Remark 1.4
• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Lemma 2.4
• proof
• Proposition 3.2
• proof