Compact Group Automorphisms, Addition Formulas and Fuglede-Kadison Determinants

By Hanfeng Li. 2010

57 citations recorded by the source corpus.

For a countable amenable group \Gamma and an element f in the integral group ring 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 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/63aa8c5f59b4db5a1608d55c6a852416c31939e9

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In Geometry and Topology

In Mathematical Physics

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.”
Fuglede–Kadison Determinants and Sofic Entropy (2014, methodology): “s such as (2). This is the ﬁrst proof of equality between Fuglede-Kadison determinants and entropy which does not use the determinant approximation. We remark that perturbations of σi(f) were used in [30] (for slightly diﬀerent purposes) in the case where Γ is amenable and λ(f) is invertible. Howeve”
ENTROPY, DETERMINANTS, AND L 2 -TORSION (2012, background): “The first author has weakened the assumptions and proved the equality between the logarithm of the Fuglede-Kadison determinant and the entropy of the associated principal algebraic action for all f ∈ ZΓ invertible in NΓ [41].”
Entropy and the variational principle for actions of sofic groups (2010, unknown): “al Xf := ZG/\ZGfvia continuous automorphisms. We may write Xf = {h∈ (R/Z)G : fh= 0}, and under this identiﬁcation αf is the restriction of the right shift action of Gon (R/Z)G to Xf (see Section 3 of [16]). In the case that f∈ ZGis invertible in ℓ1(G), Bowen showed in [4] that the soﬁc measure entro”
Homoclinic groups, IE groups, and expansive algebraic actions (2011, unknown): “D S] 2 J an 2 01”