Fuglede--Kadison determinants and entropy for actions of discrete amenable groups
By C. Deninger. 2005
78 citations recorded by the source corpus.
Consider a discrete group T and an element / in the integral group ring ZT. Then T acts from the left on the discrete additive group ZT/ZTf by automorphisms of groups. Dualizing, we obtain a left T-action on the compact Pontrjagin dual group Xf = ZT/ZTf by continuous automorphisms of groups. By definition, Xf is a closed subshift of (S1)1*. For measurable or topological actions of amenable groups a good entropy theory is available, [OW], [M]. Thus from now on let T be a finitely generated, discrete amenable group. We are interested in determining the topological entropy hf of the T-action on Xf. It agrees with the measure theoretic entropy of the T-action on Xf equipped with its Haar probability measure. In their paper [FK] from 1952, Fuglede and Kadison introduced a determinant for units in finite factor von Neumann algebras. More recently, L?ck has general ized their theory to nonunits and to group von Neumann algebras which are not necessarily factors [L2]. He needed this to define L2-torsion for general covering spaces. The von Neumann algebra A?T of the amenable group T is a certain completion of Cr and hence we may consider the Fuglede-Kadison-L?ck determinant detjvr / of/ We refer to Section 6 for the definition of a log-strong F0lner sequence. It is known that finitely generated virtually nilpotent groups have log-strong F0lner sequences. Our main result is
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/df02b5e7f8f85044da8d95d3c866c69da91acbe5
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Citations in the corpus
- Combinatorial Independence and Sofic Entropy (2012, background): “In line with previous work on algebraic actions [7, 10, 29, 34] (see also the more recent [30]), this value turns out to be equal to the logarithm of the Fuglede-Kadison determinant of f in the group von Neumann algebra LG.”
- Compact Group Automorphisms, Addition Formulas and Fuglede-Kadison Determinants (2010, background): “In [13] Deninger pointed out the possibility of h(αf) = log detLΓf for general countable amenable groups Γ and f ∈ ZΓ, and confirmed it in the special case that f is invertible in l(1)(Γ) (this is stronger than the condition that f is invertible in LΓ, see Appendix A) and positive in LΓ and that Γ h”
- Fuglede–Kadison Determinants and Sofic Entropy (2014, background): “It was Deninger in [12] who first realized that (1) has a natural generalization to noncommutative Γ via Fuglede-Kadison determinants.”
- Entropy for expansive algebraic actions of residually finite groups (2009, background): “a unital C∗ -algebra Awith respect to a tracial state tr. For any invertible fin A, its Fuglede-Kadison determinant is defined as detf := exp(trlog|f|), where |f| = (f∗f)1/2 is the absolute part of f. [De06] proved h(αf) = logdet(f) when Γ is amenable with a strong Følner sequence, fis positive in th”
- ENTROPY, DETERMINANTS, AND L 2 -TORSION (2012, methodology): “Following Deninger [19, Section 2], we interpret this result as an approximation of the Fuglede-Kadison determinant on the group von Neumann algebra of the group Z by determinants associated with Følner sets {1, . . . , n} ⊆ Z; see Example 2.2.”
- Entropy and the variational principle for actions of sofic groups (2010, unknown): “In the case of amenable acting groups and classical entropy these relationships were developed in [17, 8, 9, 16].”
- Expansive algebraic actions of discrete residually finite amenable groups and their entropy (2006, background): “This question had been left open in [3] even for the integral Heisenberg group.”
- Homoclinic groups, IE groups, and expansive algebraic actions (2011, unknown): “D S] 2 J an 2 01”