Crossed-Product von Neumann Algebras for Incompressible Navier--Stokes Flows and Spectral Complexity Indicators
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arXiv
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| Format: | Preprint |
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2026
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| author | Filardo, Gautier-Edouard Edouard |
| author_facet | Filardo, Gautier-Edouard Edouard |
| contents | 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). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_17917 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Crossed-Product von Neumann Algebras for Incompressible Navier--Stokes Flows and Spectral Complexity Indicators Filardo, Gautier-Edouard Edouard Operator Algebras Quantum Algebra Quantum Physics 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). |
| title | Crossed-Product von Neumann Algebras for Incompressible Navier--Stokes Flows and Spectral Complexity Indicators |
| topic | Operator Algebras Quantum Algebra Quantum Physics |
| url | https://arxiv.org/abs/2604.17917 |