Strong Kadison–Kastler stability and its spatial boundaries

The explanation

Take the 2-by-2 matrices that are zero off the diagonal. Call this algebra M. Rotate the plane by an angle of 0.1 radians, using a rotation matrix u. The rotated algebra N = uMu* is a tilted copy of M. The rotation moves vectors only a little: the distance from u to the identity is 2 sin(0.05), about 0.09996. The projection diag(1, 0) in M and its rotated copy in N differ by sin(0.1), about 0.0998. Every unit-size element of M has a partner in N within twice 0.09996. So a small unitary makes nearby algebras.

The Kadison–Kastler question runs the other way. Measure how far apart two algebras are by comparing their unit balls, the elements of size at most one. This Kadison–Kastler distance was introduced in 1972Source 1. If two von Neumann algebras on one space are very close, must one be a slightly rotated copy of the other? A von Neumann algebra is an algebra of operators closed under adjoints and under weak limits. Early results covered type I algebras and then amenable algebrasSource 2. Some non-amenable factors were later shown to be stable too. The obstacle was the general case: algebras of every type, with no structure to lean on.

This paper proves the uniform answer. For every ε > 0 there is a δ > 0 that works for every Hilbert space and every pair of algebras. If their distance is below δ, some unitary u with ‖u − I‖ < ε carries one onto the otherSource 3. The tolerance does not depend on the algebras, their type or their representationsSource 4.

Two companion papers mark the edges. The second looks at one-sided near inclusions, where M is only nearly inside N. It answers negatively the unrestricted question of whether small near inclusions come from small unitariesSource 5. Its examples do have unitary embeddings, but every one stays a fixed distance from the identitySource 6. The third turns to C*-algebras, which need only be closed in norm. It claims arbitrarily close separable C*-algebras that no unitary can conjugate at allSource 7. Those two algebras have the same von Neumann closureSource 7. It gives no counterexample for von Neumann algebrasSource 8.

How sure is this? A computer proof checker called Lean has a formal proof of the main theorem. It covers all unital von Neumann algebras on any complex Hilbert space, with δ depending only on ε. The two companion counterexamples are not part of that Lean statement. No specialist review of the papers has been published.

What it doesn't show. This does not extend to C*-algebras, where a companion paper claims close separable algebras need not be conjugate at all.