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

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 $\mathrm{\Phi }$ induces the Koopman unitary U on L2(M) and the crossed-product finite von Neumann algebra Mu,:= L$\mathrm{\infty }$(M) ___$\alpha$ Z= W$\star$(L$\mathrm{\infty }$(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).