Crossed-Product von Neumann Algebras for Incompressible Navier--Stokes Flows and Spectral Complexity Indicators

TL;DR

This paper develops an operator-algebraic framework to analyze incompressible flows and spectral complexity, linking tracial invariants to transport properties and enabling potential computational approaches.

Contribution

It introduces a novel von Neumann algebraic approach to quantify spectral complexity and transport in incompressible flows, connecting algebraic invariants to physical flow characteristics.

Findings

Defined tracial complexity functionals from commutators in the algebra

Connected these invariants to Fuglede--Kadison determinants and entropy-like measures

Proposed algebraic probes for transport on discretized flow benchmarks

Abstract

We introduce a traceable operator-algebraic framework for incompressible transport on M= T3 (and, more generally, compact Riemannian manifolds endowed with a smooth invariant probability measure). Given an autonomous divergence-free velocity field u, the time-1 map $\Phi$ induces the Koopman unitary U on L2(M) and the crossed-product finite von Neumann algebra Mu\,:= L$\infty$(M) ___$\alpha$ Z= W$\star$(L$\infty$(M),U), equipped with its canonical faithful normal trace $\tau$u. Within Mu we define tracial complexity functionals from commutators [U,Mf] (with Mf the multiplication operators) and associated positive elements, and we connect these quantities to Fuglede--Kadison determinants and entropy-like tracial functionals. In parallel, we introduce bounded regularized advection operators T(s) u\,:= KsTuKs as differential-level probes of transport oncommutativity, and we recall the Lie-bracket commutator identity at the formal generator level. This provides a natural algebraic setting in which tracial invariants are well posed and, in principle, computable on discretizations (e.g. cavity flow and vortex benchmarks).