NDLI: Fuglede–Kadison Determinants and Sofic Entropy

Content Provider	Springer Nature Link
Author	Hayes, Ben
Copyright Year	2016
Abstract	We relate Fuglede–Kadison determinants to entropy of finitely-presented algebraic actions in essentially complete generality. We show that if $${f\in M_{m,n}(\mathbb{Z}(\Gamma))}$$ is injective as a left multiplication operator on $${\ell^{2}(\Gamma)^{\oplus n},}$$ then the topological entropy of the action of $${\Gamma}$$ on the dual of $${\mathbb{Z}(\Gamma)^{\oplus n}/\mathbb{Z}(\Gamma)^{\oplus m}f}$$ is at most the logarithm of the positive Fuglede–Kadison determinant of f, with equality if m = n. We also prove that when m = n the measure-theoretic entropy of the action of $${\Gamma}$$ on the dual of $${\mathbb{Z}(\Gamma)^{\oplus n}/\mathbb{Z}(\Gamma)^{\oplus n}f}$$ is the logarithm of the Fuglede–Kadison determinant of f. This work completely settles the connection between entropy of principal algebraic actions and Fuglede–Kadison determinants in the generality in which dynamical entropy is defined. Our main Theorem partially generalizes results of Li-Thom from amenable groups to sofic groups. Moreover, we show that the obvious full generalization of the Li-Thom theorem for amenable groups is false for general sofic groups. Lastly, we undertake a study of when the Yuzvinskiǐ addition formula fails for a non-amenable sofic group $${\Gamma}$$ , showing it always fails if $${\Gamma}$$ contains a nonabelian free group, and relating it to the possible values of L 2-torsion in general.
Ending Page	606
Page Count	87
Starting Page	520
File Format	PDF
ISSN	1016443X
e-ISSN	14208970
Journal	Geometric and Functional Analysis
Issue Number	2
Volume Number	26
Language	English
Publisher	Springer International Publishing
Publisher Date	2016-06-13
Publisher Place	Cham
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Group algebras of locally compact groups Ergodic theory on groups Algebraic actions Analysis $C^$ -algebras and $W^$ -algebras in relation to group representations Sofic entropy General groups of measure-preserving transformations Fuglede–Kadison determinants Entropy and other invariants, isomorphism, classification
Content Type	Text
Resource Type	Article
Subject	Analysis Geometry and Topology

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