Small spatial embeddings for one-sided near inclusions of von Neumann algebras

Theorem 1.1: there are $\varepsilon_0>0$ and pairs of von Neumann algebras on separable Hilbert spaces with common identity and $\gamma(M_n,N_n)\to0$ for which spatial embeddings exist but every implementing unitary satisfies $\|u-1\|\ge\varepsilon_0$. So the unrestricted one-sided question has a negative answer, in contrast with Christensen's injective case and with the two-sided strong Kadison-Kastler theorem in the same family. The obstruction is to small unitaries, not to the existence of embeddings.

Theorem 1.1: there are $\varepsilon_0>0$ and pairs of von Neumann algebras on separable Hilbert spaces with common identity and $\gamma(M_n,N_n)\to0$ for which spatial embeddings exist but every implementing unitary satisfies $\|u-1\|\ge\varepsilon_0$. So the unrestricted one-sided question has a negative answer, in contrast with Christensen's injective case and with the two-sided strong Kadison-Kastler theorem in the same family. The obstruction is to small unitaries, not to the existence of embeddings.