Compact group automorphisms, addition formulas and Fuglede-Kadison determinants | Annals of Mathematics

Abstract For a countable amenable group Γ and an element f in the integral group ring ℤΓ being invertible in the group von Neumann algebra of Γ, we show that the entropy of the shift action of Γ on the Pontryagin dual of the quotient of ℤΓ by its left ideal generated by f is the logarithm of the Fuglede-Kadison determinant of f. For the proof, we establish an ℓp-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.

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数学年刊(Annals of Mathematics)

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