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

• URL: http://arxiv.org/abs/2604.17917v1
• Date: Mon, 20 Apr 2026 07:53:41 GMT
• Title: Crossed-Product von Neumann Algebras for Incompressible Navier--Stokes Flows and Spectral Complexity Indicators
• Abstract summary: We introduce a traceable operator-algebraic framework for incompressible transport on M= T3.<n>We define tracial complexity functionals from commutators [U,Mf] and associated positive elements.<n>We recall the Lie-bracket commutator identity at the formal generator level.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $Φ$ induces the Koopman unitary U on L2(M) and the crossed-product finite von Neumann algebra Mu\,:= L$\infty$(M) ___$α$ Z= W$\star$(L$\infty$(M),U), equipped with its canonical faithful normal trace $τ$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).

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