[Paper Review] An l^{p}-Version of von Neumann Dimension for Representations of Equivalence Relations

Ben Hayes|arXiv (Cornell University)|Feb 10, 2013

Advanced Operator Algebra Research15 references3 citations

TL;DR

This paper introduces an $l^p$-version of von Neumann dimension for representations of discrete, measure-preserving, sofic equivalence relations, extending the classical $l^2$-dimension framework. It establishes invariance under equivariant maps with dense image, computes dimensions for $L^p(\mathcal{R},\overline{\mu})^{\oplus n}$ when $1 \leq p \leq 2$, and defines an $l^p$-analogue of the first $l^2$-Betti number under finite presentation, offering new tools to probe the cost-$l^2$-Betti number conjecture.

ABSTRACT

In our previous paper, "l^{p}-Version of von Neumann Dimension for Banach Space Representations of Sofic Groups," we define an extended version of von Neumann dimension for actions of a sofic group on a Banach space. This dimension was studied especially for the translation action of G on l^{p}(G), as well as the multiplication on the non-commutative L^{p} space associated with the group von Neumann algebra. We discuss how one can similarly define an extended dimension for representations of an equivalence relation. We also define an analogue of l^{2}-Betti numbers for equivalence relations in the l^{p}-case, this may shed some light on the conjectured relation between cost and first l^{2}-Betti number.

Motivation & Objective

• To extend the theory of von Neumann dimension to $l^p$-representations of discrete, measure-preserving, sofic equivalence relations for $1 \leq p \leq 2$.
• To define an $l^p$-analogue of the first $l^2$-Betti number for equivalence relations under a finite presentation assumption.
• To investigate the relationship between this $l^p$-invariant and the conjectured equality between cost and $l^2$-Betti number (cost = $\beta_1 + 1$).
• To establish invariance of the $l^p$-dimension under equivariant maps with dense image, making it an isomorphism invariant.

Proposed method

• Adapts the methods of [13] to define an extended von Neumann dimension for $L^p$-representations of sofic equivalence relations.
• Uses direct integral decompositions and the discrete Hodge decomposition to analyze $l^p$-cohomology spaces $H_1^{(p)}(\Phi)$ associated with graphings $\Phi$.
• Applies the $\Sigma$-dimension theory for sofic equivalence relations to define $\dim_{\Sigma,l^p}$ and $\underline{\dim}_{\Sigma,l^p}$ for $\mathcal{R}$-modules.
• Employs the Dominated Convergence Theorem and measure-theoretic estimates to control convergence of $\|\delta_{\Phi_x}\zeta_x^{(n)}\|_1$ and derive contradictions in non-amenable settings.
• Leverages the structure of $L^p(E(\Phi)) / B_1^{(p)}(\Phi)$ and its decomposition into $L^p(\mathcal{R}_A, \overline{\mu}) \oplus H_1^{(p)}(\Phi_0)$ to prove lower bounds on $c_1^{(p)}$.
• Uses the fact that $L^p(\mathcal{R}, \overline{\mu})^{\oplus n}$ is isomorphic to $L^p(E(\Phi))$ for a treeing $\Phi$ of $\mathbb{F}_n$ to compute dimensions explicitly.

Experimental results

Research questions

• RQ1 Can a meaningful $l^p$-version of von Neumann dimension be defined for representations of sofic equivalence relations, generalizing the $l^2$-case?
• RQ2 Does this $l^p$-dimension remain invariant under weak isomorphism or equivariant maps with dense image?
• RQ3 What is the $l^p$-dimension of $L^p(\mathcal{R}, \overline{\mu})^{\oplus n}$ for $1 \leq p \leq 2$?
• RQ4 Can an $l^p$-analogue of the first $l^2$-Betti number be defined for equivalence relations satisfying a finite presentation condition?
• RQ5 How does the $l^p$-cohomology dimension relate to the conjectured identity $\text{cost}(\mathcal{R}) = \beta_1^{(2)}(\mathcal{R}) + 1$?

Key findings

• The $l^p$-dimension of $L^p(\mathcal{R}, \overline{\mu})^{\oplus n}$ is equal to $n$ for $1 \leq p \leq 2$, extending the classical $l^2$-dimension result.
• The $l^p$-dimension is decreasing under equivariant maps with dense image, and thus is an isomorphism invariant of the representation.
• For a free action of $\mathbb{F}_n$, the $l^p$-cohomological dimension $c_1^{(p)}(\mathcal{R})$ equals $n$, and the $l^p$-Betti number $\underline{\beta}_1^{(p)}(\Phi)$ equals $n-1$ for any graphing $\Phi$, with equality in the $\Sigma$-dimension sense.
• If $\mathcal{R}$ is hyperfinite and has infinite orbits, then $c_1^{(p)}(\mathcal{R}) = 1$, consistent with the $\mathbb{Z}$-action case.
• For any sofic approximation $\Sigma$, the inequality $c_1^{(p)}(\mathcal{R}) \leq \beta_1^{(p)}(\Phi) + 1$ holds for any graphing $\Phi$, providing a cohomological upper bound.
• The $l^p$-Betti number $\underline{\beta}_1^{(p)}(\Phi)$ is at least $n-1$ for any graphing $\Phi$ of $\mathbb{F}_n$, and equals $n-1$ when $\Phi$ is a treeing.