Abstract
For a countable amenable group $\Gamma$ and an element $f$ in the integral group ring $\mathbb{Z}\Gamma$ being invertible in the group von Neumann algebra of $\Gamma$, we show that the entropy of the shift action of $\Gamma$ on the Pontryagin dual of the quotient of $\mathbb{Z}\Gamma$ by its left ideal generated by $f$ is the logarithm of the Fuglede-Kadison determinant of $f$. For the proof, we establish an $\ell^p$-version of Rufus Bowen’s definition of topological entropy, addition formulas for group extensions of countable amenable group actions, and an approximation formula for the Fuglede-Kadison determinant of $f$ in terms of the determinants of perturbations of the compressions of $f$.
Full Article