Fuglede–Kadison determinants and entropy for actions of discrete amenable groups

Deninger, Christopher¹

¹ Mathematisches Institut, Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany

Journal of the American Mathematical Society, Tome 19 (2006) no. 3, pp. 737-758

Résumé

In 1990, Lind, Schmidt, and Ward gave a formula for the entropy of certain $\mathbb {Z}^n$-dynamical systems attached to Laurent polynomials $P$, in terms of the (logarithmic) Mahler measure of $P$. We extend the expansive case of their result to the noncommutative setting where $\mathbb {Z}^n$ gets replaced by suitable discrete amenable groups. Generalizing the Mahler measure, Fuglede–Kadison determinants from the theory of group von Neumann algebras appear in the entropy formula.

Détail

DOI : 10.1090/S0894-0347-06-00519-4

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