The hyperfinite II${}_1$-factor is Ulam stable

Abstract

We prove Ulam stability of the hyperfinite II${}_1$-factor with respect to the trace norm on the operator-norm unit ball. More precisely, every sufficiently additive, multiplicative, unital, $\ast$-preserving map from the hyperfinite II${}_1$-factor-factor into a II${}_1$-factor-factor von Neumann algebra is uniformly close, after passing to a small amplification of the target, to a genuine unital $\ast$-homomorphism. As a key finite-dimensional ingredient, we establish a dimension-free stability theorem for matrix algebras in the same trace-norm setting. As an application, we show that the hyperfinite II${}_1$-factor is isolated among II${}_1$-factors with respect to sufficiently accurate approximate $\ast$-isomorphisms.

32 pages