English

Minimal dynamical systems on prime C*-algebras

Operator Algebras 2023-12-08 v1

Abstract

We give a number of examples of exotic actions of locally compact groups on separable nuclear C*-algebras. In particular, we give examples of the following: (1) Minimal effective actions of Z{\mathbb{Z}} and FnF_n on unital nonsimple prime AF algebras. (2) For any second countable noncompact locally compact group, a minimal effective action on a separable nuclear nonsimple prime C*-algebra. (3) For any amenable second countable noncompact locally compact group, a minimal effective action on a separable nuclear nonsimple prime C*-algebra (unital when the group is Z{\mathbb{Z}} or R{\mathbb{R}}) such that the crossed product is KO2K \otimes {\mathcal{O}}_2 (O2{\mathcal{O}}_2 when the group is Z{\mathbb{Z}}). (4) For any second countable locally compact abelian group which is not discrete, an action on KO2K \otimes {\mathcal{O}}_2 such that the crossed product is a nonsimple prime C*-algebra. In most of these situations, we can specify the primitive ideal space of the C*-algebra (of the crossed product in the last item) within a class of spaces.

Keywords

Cite

@article{arxiv.2312.03980,
  title  = {Minimal dynamical systems on prime C*-algebras},
  author = {Eberhard Kirchberg and N. Christopher Phillips},
  journal= {arXiv preprint arXiv:2312.03980},
  year   = {2023}
}

Comments

This paper was nearly done about 10 years ago, but was pushed aside under the press of other projects, and then forgotten. It is now being posted, after Kirchberg's death. The key preprint of Harnisch and Kirchberg was never published and has disappeared from the website of the Universitaet Muenster. It is temporarily at: https://pages.uoregon.edu/ncp/Research/Misc/heft399.pdf

R2 v1 2026-06-28T13:43:31.814Z