Long thin covers and finite nuclear dimension for crossed products from non-free actions

Jianchao Wu  (Fudan University)

9:30-10:30úČAugust 1stúČ2024    Tian Jiabing Building 323


Finite nuclear dimension is a regularity property of C*-algebras that have played a pivotal role in the Elliott classification program of C*-algebras. It has been a key problem in the field to verify this property for crossed product C*-algebras associated to topological (as well as C*-) dynamical systems. Previous results have largely focused on the case of free actions, since existing techniques are based on Kakutani-Rokhlin-type towers in one way or another. In a recent joint work with Hirshberg, we show that any topological action by a finitely generated virtually nilpotent group on a finite-dimensional space gives rise to a crossed product C*-algebra with finite nuclear dimension. This is achieved by introducing a new topologico-dynamical dimension concept called the long thin covering dimension, which involves a suitable version of Kakutani-Rokhlin-type towers for possibly non-free actions. This result can be strengthened further and applied to some allosteric (and thus non-almost-finite) actions by certain wreath products such as the lamplighter group. Another application yields the result (joint with Eckhardt) that (twisted) group C*-algebras of virtually polycyclic groups have finite nuclear dimension.

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