Joint Normality: Uniform Perturbations and Normal Repair

We construct a jointly normal pair over a McDuff II1 factor on a fixed separable Hilbert space. Its right action admits arbitrarily small completely bounded perturbations that eliminate every nonzero spatial summand while preserving the left action and faithfulness on the minimal tensor product. The construction gives normal unital homomorphisms into a fixed II∞ factor that converge in completely bounded norm while remaining at intrinsic Bures distance √2 from their limit. This shows that the Bures–representation distance formula need not hold for general von Neumann targets. For finite targets, sufficiently small ordinary-norm perturbations of a normal unital homomorphism by normal UCP maps preserve its full reference correspondence. For finite approximately finite-dimensional sources with faithful normal tracial states, we prove uniform spatialization from normal unital positive joint maps sufficiently close on both full margins. For such sources with separable predual, a normal unital positive map on the spatial tensor product can agree exactly with both margins modulo a prescribed norm-closed range-cardinal ideal if and only if the nonspatial part has dimension below the uncountable cutoff. Uncountable homogeneous amplifications of the perturbed pair admit such a repair exactly when the cutoff equals the Hilbert space dimension and has countable cofinality. A comparison over the same factor shows that the dimension of the nonspatial part alone does not determine repair.