Close Separable C*-Algebras Without Spatial Conjugacy

We disprove the separable C${}^{\ast }$-algebraic spatial form of the Kadison–Kastler conjecture. For every ε > 0, we construct unital, norm-separable C${}^{\ast }$-algebras on a common separable complex Hilbert space, with the same identity and Kadison–Kastler distance less than ε, that are not conjugate by any unitary. The two algebras have the same von Neumann closure.