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C$^*$-diagonals in AH-algebras arising from generalized diagonal connecting maps: spectrum and uniqueness

Ali Imad Raad

TL;DR

The paper develops a Bratteli-diagram framework for AH-algebras built from generalized diagonal maps and shows that the connected components of the spectrum of the associated C$^*$-diagonal are given by inverse limits along infinite paths. It introduces spectrally incomplete components and uses them to explain non-uniqueness of inductive limit Cartan subalgebras in many AI/AH-type algebras, including Goodearl algebras and dynamical AH-models, while demonstrating that AF-algebras are spectrally complete and have unique inductive limit Cartan subalgebras. The methods connect groupoid models, eigenvalue-function data, and K-theoretic invariants to classify Cartan subalgebras in inductive limits. The results provide a clear dichotomy: spectral incompleteness yields non-uniqueness in broad AI/AH contexts, whereas AF-algebras retain a unique inductive limit Cartan structure governed by the dimension group $K_0$. Overall, the work links Bratteli-graph combinatorics with topological spectral data and standard $K$-theory to address existence and uniqueness questions for Cartan subalgebras in inductive limit C$^*$-algebras.

Abstract

We associate a Bratteli-type diagram to AH-algebras arising from generalized diagonal connecting maps. We use this diagram to give an explicit description of the connected components of the spectrum of an associated canonical C$^*$-diagonal. We introduce a topological notion on these connected components, that of being spectrally incomplete, and use it as a tool to show how various classes of AI-algebras, including certain Goodearl algebras and AH-algebra models for dynamical systems $([0,1],σ)$, do not admit unique inductive limit Cartan subalgebras. We focus on a class of spectrally complete C$^*$-algebras, namely the AF-algebras, and discuss the uniqueness of their inductive limit Cartan subalgebras.

C$^*$-diagonals in AH-algebras arising from generalized diagonal connecting maps: spectrum and uniqueness

TL;DR

The paper develops a Bratteli-diagram framework for AH-algebras built from generalized diagonal maps and shows that the connected components of the spectrum of the associated C-diagonal are given by inverse limits along infinite paths. It introduces spectrally incomplete components and uses them to explain non-uniqueness of inductive limit Cartan subalgebras in many AI/AH-type algebras, including Goodearl algebras and dynamical AH-models, while demonstrating that AF-algebras are spectrally complete and have unique inductive limit Cartan subalgebras. The methods connect groupoid models, eigenvalue-function data, and K-theoretic invariants to classify Cartan subalgebras in inductive limits. The results provide a clear dichotomy: spectral incompleteness yields non-uniqueness in broad AI/AH contexts, whereas AF-algebras retain a unique inductive limit Cartan structure governed by the dimension group . Overall, the work links Bratteli-graph combinatorics with topological spectral data and standard -theory to address existence and uniqueness questions for Cartan subalgebras in inductive limit C-algebras.

Abstract

We associate a Bratteli-type diagram to AH-algebras arising from generalized diagonal connecting maps. We use this diagram to give an explicit description of the connected components of the spectrum of an associated canonical C-diagonal. We introduce a topological notion on these connected components, that of being spectrally incomplete, and use it as a tool to show how various classes of AI-algebras, including certain Goodearl algebras and AH-algebra models for dynamical systems , do not admit unique inductive limit Cartan subalgebras. We focus on a class of spectrally complete C-algebras, namely the AF-algebras, and discuss the uniqueness of their inductive limit Cartan subalgebras.
Paper Structure (8 sections, 11 theorems, 30 equations)

This paper contains 8 sections, 11 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Groupoids and Inductive Limit Cartan Subalgebras
  4. AH-algebras from Generalized Diagonal Connecting Maps
  5. The Associated C$^*$-diagonal
  6. The Associated Bratteli Diagram and the Spectrum of the Associated C$^*$-diagonal
  7. Examples: Spectrally Incomplete AI-algebras
  8. Uniqueness of AF-Inductive Limit Cartan Subalgebras in AF-algebras

Key Result

Lemma 3.3

Let $\mathcal{Y}(n,(j,i))$ be the indexing set associated to $\phi_n^{ji}$ in equation:phiIJN. Let $p_y:(\mathcal{G}_{y},\Sigma_{y})\rightarrow (\mathcal{G}_{n}^i,\Sigma_{n}^i)$ be the groupoid model associated to the map $a\to a\circ \lambda_y$ from equation:phiIJN, $y\in \mathcal{Y}(n,(j,i))$ (c.f

Theorems & Definitions (31)

  • Definition 2.1: KumRen
  • Definition 2.2
  • Example 3.1
  • Example 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Theorem 3.5
  • proof
  • Definition 3.6
  • ...and 21 more