The strong Kadison-Kastler conjecture

Theorem 1.1: for every $\varepsilon>0$ there is $\delta(\varepsilon)>0$, independent of the algebras, their type, representations and the Hilbert space, such that unital von Neumann algebras $M,N\subseteq B(H)$ with $d(M,N)<\delta$ satisfy $uMu^*=N$ with $\|u-1\|<\varepsilon$. The proof combines amplification estimates, a modular and core analysis for type III, and reductions across types. It concerns von Neumann algebras only; the companions show that one-sided near inclusions and close separable C*-algebras behave differently. No explicit $\delta(\varepsilon)$ is given.

Theorem 1.1: for every $\varepsilon>0$ there is $\delta(\varepsilon)>0$, independent of the algebras, their type, representations and the Hilbert space, such that unital von Neumann algebras $M,N\subseteq B(H)$ with $d(M,N)<\delta$ satisfy $uMu^*=N$ with $\|u-1\|<\varepsilon$. The proof combines amplification estimates, a modular and core analysis for type III, and reductions across types. It concerns von Neumann algebras only; the companions show that one-sided near inclusions and close separable C*-algebras behave differently. No explicit $\delta(\varepsilon)$ is given.