Minicourse: Classifiability of crossed products (1)

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• Gabor Szabo (KU Leuven)
• Monday 30 June 2025, 10:30-12:00
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TGAW01 - Crossed products and groupoid C*-algebras

In this mini course we will survey the existing approaches and theorems to answer the fundamentalquestion whether a simple nuclear C-algebra arising as the crossed product of a topological dynamicalsystems falls within the scope of classification theory. As a consequence of the general structure theory fornuclear C-algebras, we now know that the key issue lies in whether such a crossed product satisfies oneof the regularity properties appearing in the Toms-Winter conjecture. Starting from a basic introductionto crossed product C-algebras, we will first set up this general question in more rigorous terms, andconsider at least two perspectives from which it can be studied. The first perspective is given by theroute of verifying that the crossed product has finite nuclear dimension, which leads us to concepts suchas Hirshberg-Winter-Zacharias’ Rokhlin dimension for group actions or Guentner-Willet-Yu’s dynamicasymptotic dimension for etale groupoids. The second perspective is given by the route of verifying that ´the crossed product is Jiang-Su stable, which leads us to concepts like Matui-Kerr’s almost finiteness forgroup actions. We shall discuss these concepts, go through a selection of the state of the art theorems,and (time permitting) take a closer look at special cases to illustrate these theorems.Recommended further reading (this will not be assumed)• For classification theory and the Toms-Winter conjecture, it can be instructive to read parts ofStrung’s book titled ”An introduction to C-algebras and the classification program” (in particular the last chapter) or the introduction of the preprint by Carrion et al titled ”Classifying -homomorphisms I: Unital simple nuclear C-algebras” (arXiv:2307.06480).• For an introduction to crossed products, groupoid C*-algebras and Rokhlin dimension, the readercan consult the book by Sims-Szabo-Williams titled ”Operator Algebras and Dynamics: Groupoids, ´Crossed products, and Rokhlin dimension”.

This talk is part of the Isaac Newton Institute Seminar Series series.

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