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arXiv · 2606.07369

The hyperfinite II$_1$-factor is Ulam stable

Vadim Alekseev·Andreas Thom
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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, $*$-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 $*$-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 $*$-isomorphisms.

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Vadim Alekseev, Andreas Thom. 2026-06-05. The hyperfinite II$_1$-factor is Ulam stable. https://arxiv.org/abs/2606.07369

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