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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 $f$.
@article{10_4007_annals_2012_176_1_5,
author = {Hanfeng Li},
title = {Compact group automorphisms, addition formulas and {Fuglede-Kadison} determinants},
journal = {Annals of mathematics},
pages = {303--347},
publisher = {mathdoc},
volume = {176},
number = {1},
year = {2012},
doi = {10.4007/annals.2012.176.1.5},
mrnumber = {2925385},
zbl = {1250.22006},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2012.176.1.5/}
}
TY - JOUR AU - Hanfeng Li TI - Compact group automorphisms, addition formulas and Fuglede-Kadison determinants JO - Annals of mathematics PY - 2012 SP - 303 EP - 347 VL - 176 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2012.176.1.5/ DO - 10.4007/annals.2012.176.1.5 LA - en ID - 10_4007_annals_2012_176_1_5 ER -
%0 Journal Article %A Hanfeng Li %T Compact group automorphisms, addition formulas and Fuglede-Kadison determinants %J Annals of mathematics %D 2012 %P 303-347 %V 176 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2012.176.1.5/ %R 10.4007/annals.2012.176.1.5 %G en %F 10_4007_annals_2012_176_1_5
Hanfeng Li. Compact group automorphisms, addition formulas and Fuglede-Kadison determinants. Annals of mathematics, Tome 176 (2012) no. 1, pp. 303-347. doi: 10.4007/annals.2012.176.1.5
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