The spatial Kadison-Kastler problem for close separable C*-algebras

Theorem 1.1: for every $\varepsilon>0$ there are unital norm-separable C*-algebras $A,B$ on a separable Hilbert space with $\mathrm{KK}(A,B)<\varepsilon$ and $A''=B''$ but no unitary with $uAu^*=B$. The construction uses sequence algebras over a deformation of the free-group regular representation (Pytlik-Szwarc family) and a tensor-norm obstruction. It does not show $A$ and $B$ are non-isomorphic, imposes no nuclearity (the CSSWW positive theorem needs it), and does not affect von Neumann algebras.

Theorem 1.1: for every $\varepsilon>0$ there are unital norm-separable C*-algebras $A,B$ on a separable Hilbert space with $\mathrm{KK}(A,B)<\varepsilon$ and $A''=B''$ but no unitary with $uAu^*=B$. The construction uses sequence algebras over a deformation of the free-group regular representation (Pytlik-Szwarc family) and a tensor-norm obstruction. It does not show $A$ and $B$ are non-isomorphic, imposes no nuclearity (the CSSWW positive theorem needs it), and does not affect von Neumann algebras.