Mean dimension, mean rank, and von Neumann–Lück rank

Abstract

We introduce an invariant, called mean rank, for any module{\mathcal{M}}of the integral group ring of a discrete amenable group Γ, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced Γ-action on the Pontryagin dual of{\mathcal{M}}, the mean rank of{\mathcal{M}}, and the von Neumann–Lück rank of{\mathcal{M}}all coincide. As applications, we establish an addition formula for mean dimension of algebraic actions, prove the analogue of the Pontryagin–Schnirelmann theorem for algebraic actions, and show that for elementary amenable groups with an upper bound on the orders of finite subgroups, algebraic actions with zero mean dimension are inverse limits of finite entropy actions.

Fuglede–Kadison Determinants and Sofic Entropy

“…Let Γ be a countable discrete sofic group with the Strong Atiyah property, and such that Proof. Our proof is essentially the same as the proof of Corollary 9.5 in [29]. It is straightforward to see that The preceding theorem and Theorem 5.1 in [19] then imply that…”

“…We may assume A = Z(Γ) ⊕n /r(f )(Z(Γ) ⊕m ) with f ∈ M m,n (Z(Γ)). By Lemma 5.4 in [29] vr(A) = dim L(Γ) (ker λ(f )), so our hypothesis implies 0 = dim L(Γ) (ker λ(f )). Thus λ(f ) is injective, so by Theorem 4.8 and so (4) is equivalent to (3).…”

“…We use mdim Σ,M , mdim Σ for metric mean dimension and mean dimension respectively (see [28] for the definition). For a Z(Γ)-module A, we use vr(A) for the von Neumann rank of A, (see [29] for the definition). Theorem 6.16.…”

Double variational principle for mean dimension

“…Since mean Hausdorff dimension is bounded between mean dimension and metric mean dimension (Proposition 3.2), we also have mdim H (X , σ, d) = prodim(X ). 19 Since r is finite, this is more restricted than in the literatures [Sch95,LL18]. They consider automorphisms of general compact Abelian groups.…”

Large dynamics of Yang–Mills theory: mean dimension formula

“…For example the Z-action on the Hilbert cube [0, 1] Z has mean dimension 1. Mean dimension has been attracting researchers in several areas such as topological dynamics [19,17,12,13,18,14], function theory [2,21,26] and operator algebra [16,7]. We review the definition of mean dimension in Section 2.1.…”