Rational $K$-Stability of Continuous $C(X)$-Algebras

TL;DR

This paper proves that rational K-stability in continuous C(X)-algebras can be transferred from fibers to the entire algebra under certain topological conditions, with applications to crossed product C*-algebras with finite Rokhlin dimension.

Contribution

It establishes the transfer of rational K-stability from fibers to the whole algebra for continuous C(X)-algebras with finite-dimensional base spaces, and applies this to crossed products with finite Rokhlin dimension.

Findings

Rational K-stability passes from fibers to the entire algebra.

Crossed product C*-algebras inherit K-stability under finite Rokhlin dimension.

Applicable to compact, metrizable, finite-dimensional base spaces.

Abstract

We show that the property of being rationally $K$-stable passes from the fibers of a continuous $C(X)$-algebra to the ambient algebra, under the assumption that the underlying space $X$ is compact, metrizable, and of finite covering dimension. As an application, we show that a crossed product C*-algebra is (rationally) $K$-stable provided the underlying C*-algebra is (rationally) $K$-stable, and the action has finite Rokhlin dimension with commuting towers.