[Paper Review] Determinants on von Neumann algebras, Mahler measures and Ljapunov exponents

Christopher Deninger|ArXiv.org|Dec 5, 2007

Advanced Operator Algebra Research8 references3 citations

TL;DR

This paper establishes a formula for the Fuglede-Kadison determinant of operators in crossed product von Neumann algebras associated with ergodic actions, expressing it in terms of Lyapunov exponents of an associated measurable cocycle. The key result generalizes Lind and Schmidt's entropy formulas for Heisenberg group actions, showing that the logarithmic determinant equals an integral over the circle of sums of positive Lyapunov exponents, with applications to noncommutative polynomials in the discrete Heisenberg group von Neumann algebra.

ABSTRACT

For an ergodic measure preserving action on a probability space, consider the corresponding crossed product von Neumann algebra. We calculate the Fuglede-Kadison determinant for a class of operators in this von Neumann algebra in terms of the Ljapunov exponents of an associated measurable cocycle. The proof is based on recent work of Dykema and Schultz. As an application one obtains formulas for the Fuglede-Kadison determinant of noncommutative polynomials in the von Neumann algebra of the discrete Heisenberg group. These had been previously obtained by Lind and Schmidt via entropy considerations.

Motivation & Objective

• To calculate the Fuglede-Kadison determinant for a class of operators in crossed product von Neumann algebras arising from ergodic measure-preserving actions.
• To relate the determinant to Lyapunov exponents of a measurable cocycle via the Oseledets multiplicative ergodic theorem.
• To provide a functional-analytic derivation of Lind and Schmidt's entropy formulas for algebraic actions of the discrete Heisenberg group.
• To extend the determinant formula to noncommutative polynomials in the group von Neumann algebra of the Heisenberg group.

Proposed method

• Uses the Fuglede-Kadison determinant in finite von Neumann algebras with a faithful normal finite trace.
• Applies the Oseledets multiplicative ergodic theorem to a matrix cocycle $ A_{ ho} $ associated with the operator $ ho $.
• Employs Dykema and Schultz's work on the Aluthge transform and Brown measure to relate the determinant to Lyapunov exponents.
• Reduces the general case to the continuous case via Margulis' method, leveraging uniform convergence in uniquely ergodic systems.
• Uses direct integral decomposition of the von Neumann algebra $ \mathcal{N}\Gamma $ of the discrete Heisenberg group into rotation algebras $ \mathcal{R}_\zeta $.
• Applies the trace formula $ \log\det_{\mathcal{M}}\Phi = \int_Z \log\det_{\mathcal{M}_\zeta}\Phi_\zeta \, d\mu(\zeta) $ for decomposable operators.

Experimental results

Research questions

• RQ1 How can the Fuglede-Kadison determinant of a noncommutative polynomial in the discrete Heisenberg group von Neumann algebra be computed?
• RQ2 What is the precise relationship between Lyapunov exponents of a measurable cocycle and the Fuglede-Kadison determinant in crossed product von Neumann algebras?
• RQ3 Can the entropy formulas of Lind and Schmidt for algebraic $ \Gamma $-actions be derived from von Neumann algebra techniques?
• RQ4 Under what conditions does the logarithmic determinant equal the integral of the positive part of the Lyapunov exponent?
• RQ5 How does the determinant behave under direct integral decompositions of von Neumann algebras?

Key findings

• For $ \Phi = 1 - a(y,z)x $ in the complex group ring of the discrete Heisenberg group, the logarithmic Fuglede-Kadison determinant is $ \int_{S^1} \left( \int_{S^1} \log|a(\eta,\zeta)| \, d\mu(\eta) \right)^+ \, d\mu(\zeta) $.
• For general $ \Phi = \sum_{i=0}^N a_i(y,z)x^i $, the logarithmic determinant is $ \int_{S^1} \sum_j r_j(\zeta) \chi_j(\zeta)^+ \, d\mu(\zeta) $, where $ \chi_j(\zeta) $ are the Lyapunov exponents of the matrix $ A_{\Phi_\zeta} $ with multiplicities $ r_j(\zeta) $.
• The formula holds $ \mu $-almost everywhere on $ S^1 $, with $ \log|a(\cdot,\zeta)| \in L^1(S^1) $ for $ \mu $-a.e. $ \zeta $, ensuring integrability.
• The result extends to measurable coefficient functions $ a_i $, not just continuous ones, via reduction to the continuous case using ergodic theory techniques.
• The determinant formula is consistent with entropy calculations: $ \log\det_{\mathcal{N}\Gamma}\Phi = h_\Phi $, the entropy of the $ \Gamma $-action on the Pontrjagin dual of $ \mathbb{Z}\Gamma / \mathbb{Z}\Gamma\Phi $.
• The unitarity of $ \Phi_\zeta $ in $ \mathcal{R}_\zeta $ is equivalent to $ \int_{S^1} \log|a(\eta,\zeta)| \, d\mu(\eta) \neq 0 $ for $ \mu $-a.e. $ \zeta $.