Compact Group Automorphisms, Addition Formulas and Fuglede-Kadison Determinants
arXiv:1001.0419
Abstract
For a countable amenable group Γand an element f in the integral group ring ZΓbeing invertible in the group von Neumann algebra of Γ, we show that the entropy of the shift action of Γon the Pontryagin dual of the quotient of ZΓby 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.
To appear in Ann. of Math., 41 pages
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