Existence of large-n limits of sentences evaluated in matrix algebras

Determine whether, for every sentence φ in the language of tracial von Neumann algebras, the sequence φ^{M_n} (with M_n the n×n matrix algebra equipped with the normalized trace) converges as n → ∞.

Background

A core technical ingredient of the paper is to work with formulas (sentences when there are no free variables) in continuous logic for tracial von Neumann algebras. Understanding the large-n behavior of these formula evaluations on matrix algebras is crucial for connecting model-theoretic type limits to random matrix limits.

While ultraproduct limits along ultrafilters exist and are used in the paper, the authors note that it is unknown whether the straight limit as n → ∞ exists for general sentences; related negative phenomena are known in analogous settings (e.g., permutation groups with the Hamming metric). Establishing existence would have broad implications, including progress toward large deviations programs for invariant ensembles.