[Paper Review] Approximation of center-valued Betti-numbers

Anselm Knebusch|ArXiv.org|Apr 3, 2008

Advanced Operator Algebra Research7 references3 citations

TL;DR

This paper generalizes Lück's approximation theorem for $L^2$-Betti numbers to center-valued Betti numbers, which are finer invariants derived from the universal trace on the group von Neumann algebra. It establishes that the Fourier coefficients of center-valued Betti numbers—related to delocalized Betti numbers—can be approximated via finite quotients of the fundamental group, extending the approximation framework to non-trace-valued dimensions in non-commutative geometry.

ABSTRACT

In this paper we generalize the approximation theorem for L^2-Betti numbers to an approximation theorem for center-valued Betti-numbers.

Motivation & Objective

• To extend the classical approximation theorem for $L^2$-Betti numbers to the more refined center-valued Betti numbers, which measure homology dimensions using the center-valued trace on the group von Neumann algebra.
• To establish that the Fourier coefficients of center-valued Betti numbers—linked to delocalized Betti numbers—can be approximated by their finite-dimensional counterparts in finite quotients of the fundamental group.
• To generalize the approximation framework to groups in the class $\mathcal{G}$, including residually finite and amenable groups, using trace approximation techniques on matrix algebras over group rings.
• To provide a technical foundation for the Atiyah conjecture and related invariants by extending approximation results beyond complex-valued traces to universal traces in finite von Neumann algebras.

Proposed method

• Uses the universal (center-valued) trace $\operatorname{tr}^u_{\mathcal{N}(G)}$ on the group von Neumann algebra $\mathcal{N}(G)$ to define center-valued Betti numbers as $\dim^u_{\mathcal{N}(G)}(\ker \Delta_p)$.
• Applies the delocalized trace $\operatorname{tr}^{\langle g\rangle}_{\mathcal{N}(G)}$ for elements $g$ with finite conjugacy class to extract Fourier coefficients of the center-valued dimension.
• Relies on approximation of spectral projections and distribution functions $F^{\langle g\rangle,\operatorname{Re}}_A(\lambda)$ associated with matrices $A \in M_d(\mathbb{Z}G)$ under group quotient maps $G_i \to G$.
• Employs the logarithmic integral $\int_{0^+}^{\kappa(A)} \frac{F^{\langle g\rangle,\operatorname{Re}}_A(\lambda) - F^{\langle g\rangle,\operatorname{Re}}_A(0)}{\lambda} d\lambda$ as a proxy for the trace of the logarithm of operators.
• Uses uniform bounds on the real part of the spectral distribution functions and their limits to prove convergence of the delocalized traces under the quotient process.
• Establishes convergence of the delocalized trace $\operatorname{tr}^{\langle g\rangle}_{\mathcal{N}(G_i)}(A[i])$ to $\operatorname{tr}^{\langle g\rangle}_{\mathcal{N}(G)}(A)$ via limit inferior and superior arguments on distribution functions.

Experimental results

Research questions

• RQ1 Can the approximation theorem for $L^2$-Betti numbers be extended to center-valued Betti numbers, which are more refined invariants in non-commutative geometry?
• RQ2 Do the Fourier coefficients of center-valued Betti numbers—corresponding to delocalized Betti numbers—admit approximation via finite quotients of the fundamental group?
• RQ3 Is the convergence of the delocalized trace $\operatorname{tr}^{\langle g\rangle}_{\mathcal{N}(G_i)}(A[i])$ to $\operatorname{tr}^{\langle g\rangle}_{\mathcal{N}(G)}(A)$ uniform and stable under the quotient process for groups in $\mathcal{G}$?
• RQ4 Can the approximation framework be extended beyond complex-valued traces to universal traces in finite von Neumann algebras?

Key findings

• The center-valued Betti numbers $\beta_p^u(X)$ can be approximated via the finite-dimensional analogs $\beta_p^u(\widetilde{X}/\pi_i)$, extending Lück’s theorem to the universal trace setting.
• The Fourier coefficients of the center-valued dimension, which are multiples of delocalized Betti numbers, converge under the quotient process $G_i \to G$.
• The limit of the delocalized trace $\operatorname{tr}^{\langle g\rangle}_{\mathcal{N}(G_i)}(A[i])$ exists and equals $\operatorname{tr}^{\langle g\rangle}_{\mathcal{N}(G)}(A)$, as shown by convergence of spectral distribution functions.
• The approximation holds uniformly for matrices $A \in M_d(\mathbb{Z}G)$ over groups in the class $\mathcal{G}$, including residually finite and amenable groups.
• The key technical step is proving that $\liminf_{i\in I} F^{\langle g\rangle,\operatorname{Re}}_{A[i]}(0) = \limsup_{i\in J} F^{\langle g\rangle,\operatorname{Re}}_{A[i]}(0) = F^{\langle g\rangle,\operatorname{Re}}_A(0)$, ensuring trace convergence.
• The proof relies on bounding the logarithmic integral of the spectral distribution difference, showing it is uniformly bounded by $2d \ln(\kappa(A))$, which implies convergence of the traces.