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

Deninger, Christopher

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

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

J. Amer. Math. Soc. 19 (2006), 737-758

DOI: https://doi.org/10.1090/S0894-0347-06-00519-4

Published electronically: February 6, 2006

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Abstract:

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.

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