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$G$-kernels of Kirchberg algebras

Masaki Izumi

TL;DR

The paper advances the study of G-kernels α:G→Out(A) by introducing a K-theory–driven invariant tilde{ob}(α)∈H^3(G,K^#_0(A)) that refines the classical ob(α)∈H^3(G,𝕋) and remains informative in purely infinite settings. It develops a cocycle-action invariant κ^3(α,u) and relates tilde{ob}(α) to ob(α) via ev_1, while exploiting continuous-field (Dadarlat–Pennig) theory to classify Z^n-kernels for strongly self-absorbing Kirchberg algebras in the bootstrap category. By constructing semigroups F_A(G) and E_A(G) of G-kernel and cocycle-action classes and connecting them to DP theory, the work provides a cohesive framework for understanding realization, classification, and structure of G-kernels in this regime, with concrete results for A in D_{pi} and amenable G (e.g., Z^n, poly-Z). The results illuminate how K-theory–based obstructions interact with continuous-field techniques to yield both local and global classification information, and they propose concrete conjectures and partial results for when these semigroups form groups and how obstructions realize in various Kirchberg settings.

Abstract

A $G$-kernel is a group homomorphism from a group $G$ to the outer automorphism group of a C$^*$-algebra. Inspired by recent work of Evington and Girón Pacheco in the stably finite case, we introduce a new invariant of a $G$-kernel using $K$-theory, and deduce several new constraints of the obstruction classes of $G$-kernels in the purely infinite case. We classify $\mathbb{Z}^n$-kernels for strongly self-absorbing Kirchberg algebras in the bootstrap category in terms of our new invariant and the Dadarlat-Pennig theory of continuous fields of strongly self-absorbing C$^*$-algebras.

$G$-kernels of Kirchberg algebras

TL;DR

The paper advances the study of G-kernels α:G→Out(A) by introducing a K-theory–driven invariant tilde{ob}(α)∈H^3(G,K^#_0(A)) that refines the classical ob(α)∈H^3(G,𝕋) and remains informative in purely infinite settings. It develops a cocycle-action invariant κ^3(α,u) and relates tilde{ob}(α) to ob(α) via ev_1, while exploiting continuous-field (Dadarlat–Pennig) theory to classify Z^n-kernels for strongly self-absorbing Kirchberg algebras in the bootstrap category. By constructing semigroups F_A(G) and E_A(G) of G-kernel and cocycle-action classes and connecting them to DP theory, the work provides a cohesive framework for understanding realization, classification, and structure of G-kernels in this regime, with concrete results for A in D_{pi} and amenable G (e.g., Z^n, poly-Z). The results illuminate how K-theory–based obstructions interact with continuous-field techniques to yield both local and global classification information, and they propose concrete conjectures and partial results for when these semigroups form groups and how obstructions realize in various Kirchberg settings.

Abstract

A -kernel is a group homomorphism from a group to the outer automorphism group of a C-algebra. Inspired by recent work of Evington and Girón Pacheco in the stably finite case, we introduce a new invariant of a -kernel using -theory, and deduce several new constraints of the obstruction classes of -kernels in the purely infinite case. We classify -kernels for strongly self-absorbing Kirchberg algebras in the bootstrap category in terms of our new invariant and the Dadarlat-Pennig theory of continuous fields of strongly self-absorbing C-algebras.
Paper Structure (12 sections, 28 theorems, 145 equations, 6 figures)

This paper contains 12 sections, 28 theorems, 145 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A new invariant
  4. Invariant $\kappa^3(\alpha,u)$ for a cocycle action $(\alpha,u)$.
  5. Invariant $\operatorname{\widetilde{ob}}(\alpha)$ for a $G$-kernel $\alpha$.
  6. Stably finite case
  7. Strongly self-absorbing Kirchberg algebras
  8. Semigroup $\mathcal{F}_A(G)$
  9. Semigroup $\mathcal{E}_A(G)$
  10. Relationship with the Dadarlat-Pennig theory
  11. Structure of $\mathcal{F}_A(G)$
  12. Comments

Key Result

Theorem 2.2

Let $G$ be a countable amenable group, and let $\mathcal{R}$ be the hyperfinite II$_1$ factor. Then the third cohomology obstruction is a completely invariant for $G$-kernels up to conjugacy. Moreover, every class in $H^3(G,\mathbb{T})$ can be realized in $\mathcal{R}$.

Figures (6)

  • Figure 1: $(e:\Delta^3:g_1,g_2,g_3)$
  • Figure 2: $(e:\Delta^2:g_1,g_2)$
  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 1 more figures

Theorems & Definitions (59)

  • Definition 2.1
  • Theorem 2.2: Connes, Jones, Ocneanu
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Theorem 2.5: EGP23
  • Theorem 2.6: EGP23
  • Definition 2.7
  • Remark 2.8
  • Conjecture 2.9
  • ...and 49 more