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Remarks on Villadsen algebras, II: A generalized construction and the comparison radius function

George A. Elliott, Zhuang Niu

TL;DR

The paper broadens Villadsen algebra classification from fixed seed spaces to AF-Villadsen algebras by introducing a refined invariant, the comparison radius function $r^{(0)}_ exists$, which lives on the $K_0$-level and captures dimension-growth data from Bratteli diagrams and point evaluations. It proves an isomorphism theorem showing AF-Villadsen algebras are classified by their $K_0$-group together with $r^{(0)}_ exists$, equivalently by the Cuntz semigroup, under rapid-growth assumptions. A generalized radius function $r_ullet$ for UHF-Villadsen algebras is developed, linking it to the radius of comparison and enabling a nuanced analysis of trace spaces and automorphism actions. The work demonstrates seed-space differences can be detected via Cuntz semigroup data, and provides examples where $r_ullet$ is not constant, affecting classification and symmetry properties. Overall, it deepens the connection between Bratteli-constructed inductive limits, radius-type invariants, and the Cuntz semigroup in the classification of Villadsen-type algebras.

Abstract

The authors' recent classification of Jesper Villadsen's remarkable generalization (based on a self-reproducing seed space) of Glimm's infinite tensor product (UHF) C*-algebras, by means of the Cuntz semigroup (in the case of a fixed, well-behaved, seed space), is extended to the analogous generalization of Bratteli's approximately finite-dimensional (AF) C*-algebras. Some progress is made in the direction of distinguishing between algebras based on different seed spaces.

Remarks on Villadsen algebras, II: A generalized construction and the comparison radius function

TL;DR

The paper broadens Villadsen algebra classification from fixed seed spaces to AF-Villadsen algebras by introducing a refined invariant, the comparison radius function , which lives on the -level and captures dimension-growth data from Bratteli diagrams and point evaluations. It proves an isomorphism theorem showing AF-Villadsen algebras are classified by their -group together with , equivalently by the Cuntz semigroup, under rapid-growth assumptions. A generalized radius function for UHF-Villadsen algebras is developed, linking it to the radius of comparison and enabling a nuanced analysis of trace spaces and automorphism actions. The work demonstrates seed-space differences can be detected via Cuntz semigroup data, and provides examples where is not constant, affecting classification and symmetry properties. Overall, it deepens the connection between Bratteli-constructed inductive limits, radius-type invariants, and the Cuntz semigroup in the classification of Villadsen-type algebras.

Abstract

The authors' recent classification of Jesper Villadsen's remarkable generalization (based on a self-reproducing seed space) of Glimm's infinite tensor product (UHF) C*-algebras, by means of the Cuntz semigroup (in the case of a fixed, well-behaved, seed space), is extended to the analogous generalization of Bratteli's approximately finite-dimensional (AF) C*-algebras. Some progress is made in the direction of distinguishing between algebras based on different seed spaces.
Paper Structure (7 sections, 20 theorems, 317 equations)

This paper contains 7 sections, 20 theorems, 317 equations.

Table of Contents

  1. Introduction
  2. AF-Villadsen algebras and the function $r_\infty^{(0)}$
  3. Growth along a Bratteli diagram
  4. Adding point evaluations and the function $r_\infty^{(0)}$
  5. The comparison property of $r^{(0)}_\infty$
  6. The isomorphism theorem
  7. A generalized version of the comparison radius function for UHF-Villadsen algebras

Key Result

Theorem 1.1

Let $X$ be a K-contractible solid space such that $0 < \mathrm{dim}(X) < \infty$, and let $A(X, G, \mathcal{E})$ and $B(X, H, \mathcal{F})$ be AF-Villadsen algebras with seed space $X$ with rapid dimension growth (see rapid-growth-cond), where $G$ and $H$ are Bratteli diagrams and $\mathcal{E}$ and where $r_\infty^{(0)}{(A)}$ and $r_\infty^{(0)}{(B)}$ are the comparison radius functions of $A$ an

Theorems & Definitions (52)

  • Theorem 1.1: Theorem \ref{['classification-AF']}
  • Theorem 1.2: Theorem \ref{['low-env-gn']}
  • Theorem 1.3: Corollary \ref{['joint-classification']}
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • ...and 42 more