Journal of Operator Theory

Volume 95, Issue 1, Winter 2026 pp. 247-274.

Continuity and equivariant dimension

Authors: Alexandru Chirvasitu (1), Benjamin Passer (2)
Author institution: (1) Department of Mathematics, University at Buffalo, Buffalo, NY, 14260, U.S.A.
(2) Department of Mathematics, United States Naval Academy, Annapolis, MD, 21402, U.S.A.

Summary: We study local-triviality dimensions of actions on $C^*$-algebras, which are 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 vary non-continuously. 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.

DOI: http://dx.doi.org/10.7900/jot.2024may17.247
Keywords: local-triviality dimension, continuous fields, deformations, free actions, noncommutative sphere, noncommutative torus, vector bundle, matrix bundle

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