Continuous coefficients and von Neumann closure [connes-000E]

✍️source

continuousCrossedClosure_mem_iff

L∃∀NAGENTDRAFTED

Coordinate characters have norm-dense linear span in the continuous functions by Stone–Weierstrass. Since continuous coefficients act continuously as crossed-product multipliers, the generic continuous-coefficient closure theorem Connes.CrossedProduct.vonNeumannClosure_crossedGeneratorSetOfContinuousCoefficients_eq upgrades that density to equality of the generated von Neumann closures.

Pullback along the homeomorphism then gives Connes.CrossedProduct.continuousCrossedClosure_mem_iff. This does not say that continuous functions are norm-dense in \(L^\infty \); it compares the von Neumann closures generated after adjoining all continuous coefficients.

The paper-facing proof applies the transport and closure bridges once to each action, then composes the Fourier models with the shear. This removes parallel action-one/action-two closure plumbing while preserving Zhou's Proposition 3.4 endpoint and section order. Mapping a convenient generating family remains an intermediate step; the consumer-facing contract is equivalence of membership in the completed von Neumann algebras.