Continuity and equivariant dimension
Abstract
We study the local-triviality dimensions of actions on -algebras, which are invariants developed for noncommutative Borsuk-Ulam theory. While finiteness of the local-triviality dimensions is known to guarantee freeness of an action, we show that free actions need not have finite weak local-triviality dimension. Moreover, the local-triviality dimensions of a continuous field may be greater than those of its individual fibers, and the dimensions may fail to vary continuously across the fibers. However, in certain circumstances upper semicontinuity of the weak local-triviality dimension is guaranteed. We examine these results and counterexamples with a focus on noncommutative tori and noncommutative spheres, both in terms of computation and theory.
Keywords
Cite
@article{arxiv.2403.06767,
title = {Continuity and equivariant dimension},
author = {Alexandru Chirvasitu and Benjamin Passer},
journal= {arXiv preprint arXiv:2403.06767},
year = {2026}
}
Comments
v4 makes small corrections to Proposition 4.5 and removes Proposition 4.9; to appear in the Journal of Operator Theory; 21 pages + references