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$G$-kernels of Kirchberg algebras

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arxiv 2309.03441 v3 pith:QW45HJUZ submitted 2023-09-07 math.OA

classification math.OA
keywords algebrasgroupkernelscaseinvariantkernelkirchbergself-absorbing
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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 Gir\'{o}n 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.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Classification of anomalous actions of finite groups with the Rokhlin property

    math.OA 2026-08 accept novelty 8.0 of 10

    Rokhlin G-kernels on Kirchberg algebras are classified by their lifting obstruction in H^3(G,T) and the induced G-module structure on K-theory.

  2. G-kernels and Crossed Modules

    math.OA 2025-09 conditional novelty 7.0 of 10

    Topological crossed modules unify cohomological invariants of G-kernels; for strongly self-absorbing algebras the crossed-module classifying space is equivalent to the bundle-theoretic one and a restricted cohomology ...

  3. Infinite Loop Spaces and Group Actions on Strongly Self-Absorbing C*-Algebras

    math.AT 2026-07 accept novelty 6.0 of 10

    For strongly self-absorbing C*-algebras, the classifying spaces for cocycle actions and Γ-kernels are infinite loop spaces, so H^1 obstruction sets take values in cohomology groups.

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