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$C^*$-diagonal of inductive limit of $1$-dimensional NCCW complexes

Dolapo Oyetunbi

TL;DR

This work proves the existence of a $C^*$-diagonal for unital inductive limits of $1$-dimensional NCCW complexes with trivial $K_1$-groups and injective connecting maps. It develops $n$-standard maps to approximate arbitrary unital $^*$-homomorphisms between such complexes, and introduces a $ ext{D}$-pair structure to realize these maps via piecewise unitary conjugations, extending ideas from $1$-dimensional CW complexes to NCCW settings. By constructing diagonals on the building blocks and ensuring their compatibility through the inductive system, the paper shows that the limit carries a Cartan subalgebra with the unique extension property. This extends the landscape of $C^*$-diagonal results beyond simple AH-algebras and connects detailed invariants and standard-map techniques to the diagonalization process in non-simple, NCCW-based inductive limits.

Abstract

This paper establishes the existence of a $C^*$-diagonal in the inductive limit of 1-dimensional NCCW complexes with trivial $K_1$-groups. It also examines some limitations and implications of approximating $^*$-homomorphisms between two such complexes.

$C^*$-diagonal of inductive limit of $1$-dimensional NCCW complexes

TL;DR

This work proves the existence of a -diagonal for unital inductive limits of -dimensional NCCW complexes with trivial -groups and injective connecting maps. It develops -standard maps to approximate arbitrary unital -homomorphisms between such complexes, and introduces a -pair structure to realize these maps via piecewise unitary conjugations, extending ideas from -dimensional CW complexes to NCCW settings. By constructing diagonals on the building blocks and ensuring their compatibility through the inductive system, the paper shows that the limit carries a Cartan subalgebra with the unique extension property. This extends the landscape of -diagonal results beyond simple AH-algebras and connects detailed invariants and standard-map techniques to the diagonalization process in non-simple, NCCW-based inductive limits.

Abstract

This paper establishes the existence of a -diagonal in the inductive limit of 1-dimensional NCCW complexes with trivial -groups. It also examines some limitations and implications of approximating -homomorphisms between two such complexes.
Paper Structure (4 sections, 15 theorems, 71 equations)

This paper contains 4 sections, 15 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maximally Homogeneity and Standard Maps
  4. $C^*$-diagonals of Inductive Limits of $1$-dimensional NCCW complexes

Key Result

Theorem 1.1

Every unital inductive limit of one-dimensional NCCW complexes with trivial $K_1$-group and unital injective connecting maps has a $C^*$-diagonal.

Theorems & Definitions (39)

  • Theorem 1.1: see Theorem \ref{['thm:Mainresult']}
  • Corollary 1.2: see Corollary \ref{['cor:InductiveLimit']}
  • Definition 2.1
  • Theorem 2.2: APS11
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Proposition 2.6
  • Lemma 3.1: Liu19b
  • Definition 3.2
  • ...and 29 more