Covering entropy for types in tracial W<sup>*</sup>-algebras

Abstract

We study embeddings of tracial $\mathrm{W}^*$-algebras into a ultraproduct of matrix algebras through an amalgamation of free probabilistic and model-theoretic techniques. Jung implicitly and Hayes explicitly defined \emph{$1$-bounded entropy} through the asymptotic covering numbers of Voiculescu's microstate spaces, that is, spaces of matrix tuples $(X_1^{(N)},X_2^{(N)},\dots)$ having approximately the same $*$-moments as the generators $(X_1,X_2,\dots)$ of a given tracial $\mathrm{W}^*$-algebra. We study the analogous covering entropy for microstate spaces defined through formulas that use suprema and infima, not only $*$-algebra operations and the trace | formulas such as arise in the model theory of tracial $\mathrm{W}^*$-algebras initiated by Farah, Hart, and Sherman. By relating the new theory with the original $1$-bounded entropy, we show that if $\mathcal{M}$ is a separable tracial $\mathrm{W}^*$-algebra with $h(\cN:\cM) \geq 0$, then there exists an embedding of $\cM$ into a matrix ultraproduct $\cQ = \prod_{n \to \cU} M_n(\C)$ such that $h(\cN:\cQ)$ is arbitrarily close to $h(\cN:\cM)$. We deduce that if all embeddings of $\cM$ into $\cQ$ are automorphically equivalent, then $\cM$ is strongly $1$-bounded and in fact has $h(\cM) \leq 0$.