Measure-free Koopman-von Neumann Dynamics and Noncommutative Geometry

What the math gives to ML

The paper provides a constructive dilation of a generally non-skew Koopman generator into a skew-adjoint operator on an enlarged Hilbert space. Writing the generator as V=S+A, with S=(V+V*)/2 self-adjoint and A=(V-V*)/2 skew-adjoint, it embeds V into a block-tridiagonal skew operator whose unitary evolution is stable in the lifted norm while its projection recovers the original non-unitary dynamics on analytic observables. This mechanism transfers naturally to recurrent or state-space neural networks: represent hidden states in a multi-channel latent space, constrain the lifted generator to be skew-adjoint, and decode only a projected component. A Cayley time discretization preserves the lifted norm exactly, giving a falsifiable stability signature while still permitting non-normal, dissipative, or amplifying effective dynamics after projection.