Compact group automorphisms, addition formulas and Fuglede-Kadison determinants. (English) Zbl 1250.22006

Li, Hanfeng

Ann. Math. (2) 176, No. 1, 303-347 (2012).

Dynamical systems of algebraic origin give the simplest and most homogeneous examples, and as a result it is useful to be able to compute dynamical invariants in this setting explicitly. The topological entropy of automorphisms of compact groups was computed by S. A. Yuzvinskii [Sib. Math. J. 8, 172–178 (1967); translation from Sib. Mat. Zh. 8, 230–239 (1967; Zbl 0211.34901)], and this was later extended to an expression for the entropy of a \(\mathbb Z^d\)-action by automorphisms of a compact group in terms of the Mahler measure of associated polynomials in work of D. Lind, K. Schmidt and the reviewer [Invent. Math. 101, No. 3, 593–629 (1990; Zbl 0774.22002)]. Dealing with a non-abelian compact group presents no serious obstacles, but extending this result from actions of \(\mathbb Z^d\) to actions of other amenable groups presents formidable difficulties. A key breakthrough was made by C. Deninger [J. Am. Math. Soc. 19, No. 3, 737–758 (2006; Zbl 1104.22010)], who showed that the Fuglede-Kadison determinant [B. Fuglede and R. V. Kadison, Ann. Math. (2) 55, 520–530 (1952; Zbl 0046.33604)] could be used to express the entropy of certain algebraic actions of amenable groups. This was generalized by C. Deninger and K. Schmidt [Ergodic Theory Dyn. Syst. 27, No. 3, 769–786 (2007; Zbl 1128.22003)], but in all cases the results required conditions (coming from properties of the defining polynomial in the associated von Neumann algebra) on the action which were restrictive and opaque. Here a new approach is found, building on work of L. Bowen [Ergodic Theory Dyn. Syst. 31, No. 3, 703–718 (2011; Zbl 1234.37010)], and giving a complete result in the most natural case. To state this, let \(G\) be a countable amenable group and fix \(f\) in the integral group ring \(\mathbb Z G\) of \(G\). Then the entropy of the natural shift \(G\)-action on the character group of the quotient of \(\mathbb Z G\) by the left ideal generated by \(f\) is equal to the logarithm of the Fuglede-Kadison determinant of \(f\). This is a significant step towards an understanding of the dynamics of these basic building blocks of algebraic actions of amenable group actions.

Reviewer: Thomas B. Ward (Durham)

MSC:

22D25	\(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
37B40	Topological entropy
37A35	Entropy and other invariants, isomorphism, classification in ergodic theory
28D05	Measure-preserving transformations

Keywords:

entropy; algebraic dynamics; Fuglede-Kadison determinant

Citations:

Zbl 0211.34901; Zbl 0774.22002; Zbl 1104.22010; Zbl 0046.33604; Zbl 1128.22003; Zbl 1234.37010

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