A dynamical approach to von Neumann dimension

Abstract

Let Γ be an amenable group and V be a finite dimensional vector space. Gromov pointed out that the von Neumann dimension of linear subspaces of ℓ 2 (Γ; V ) (with respect to Γ) can be obtained by looking at a growth factor for a dynamical (pseudo-)distance. This dynamical point of view (reminiscent of metric entropy) does not requires a Hilbertian structure. It is used in this article to associate to a Γ-invariant linear subspaces Y of ℓ p (Γ; V ) a real positive number dim ℓ p Y (which is the von Neumann dimension when p = 2). By analogy with von Neumann dimension, the properties of this quantity are explored to conclude that there can be no injective Γ-equivariant linear map of finite-type fromHence, we are looking for a notion of dimension for such subspaces, which would increase under injective equivariant linear maps. Inspired by an argument of [6, §1.12] (and partially answering the question found therein), we shall introduce a quantity dim ℓ p which, when p = 2, coincides with definition of von Neumann dimension. This quantity is obtained by a process similar to that of metric entropy or mean dimension, i.e. by looking at an asymptotic growth factor. The definition relies a priori on an exhaustion of Γ, but a generalization of the Ornstein-Weiss lemma in section 5 implies the result is independent of this choice.Though we prove many properties of dim ℓ p , important properties are still lacking. Nevertheless, the results obtained in this paper suffice to establish a non existence result for maps of finite type. We recall their construction.Let D ⊂ Γ be a finite set and let g : V D → V ′ be a continuous map. This data enables the definition of a Γ-equivariant continuous map gRemark that what we denote here as ℓ p (Γ; V ) is more frequently written ℓ p (Γ) ⊗ V .Theorem 1.1. Let Γ be an amenable discrete group. Let V and V ′ be finite dimensional vector spaces. If f :Consequently, if we restrict ourselves to maps of finite type, the question above has a positive answer: there is a Γ-isomorphism of finite type between ℓ p (Γ; R n ) and ℓ p (Γ; R m ) if and only if m = n.

An obstruction to \ell ^{p}-dimension

“…This has prompted recent work in pursuit of a more general p -dimension for 1 < p < ∞; see e.g. [9,8,5,6]. The purpose of this short note is to establish a fundamental obstruction to this endeavour.…”

“…The proof that we can take the kernel of T as our F is analogous to the proof of Proposition 0.5. (v) Using an Euler characteristic argument, Gaboriau had previously observed that for certain non-amenable groups G one cannot hope to have a notion of p -dimension for which ( p G) ⊕n has dimension n, and which also satisfies additivity for short exact sequences (see the introduction of [5]). The fact that we produce thin exhaustions for amenable groups should further increase the doubts that there be any reasonable p -dimension for large p.…”

Instanton approximation, periodic ASD connections, and mean dimension

“…For the detail, see Gromov [14] and Lindenstrauss and Weiss [16]. For some related works, see Lindenstrauss [15] and Gournay [10][11][12][13].…”

An lp-version of von Neumann dimension for Banach space representations of sofic groups II

“…These definitions may seem quite technical and bizarre, but they are really inspired by ideas of Bowen [2], Kerr and Li [18], Gournay [10] and Voiculescu [29].…”