Covering entropy for types in tracial ${\mathrm{W}}^{\ast }$-algebras

Abstract

We study embeddings of tracial ${\mathrm{W}}^{\ast }$-algebras into a ultraproduct of matrix algebras through an amalgamation of free probabilistic and model-theoretic techniques. Jung implicitly and Hayes explicitly defined $1$-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 $\ast$-moments as the generators $(X_1,X_2,\dots )$ of a given tracial ${\mathrm{W}}^{\ast }$-algebra. We study the analogous covering entropy for microstate spaces defined through formulas that use not only $\ast$-algebra operations and the trace, but also suprema and infima, such as arise in the model theory of tracial ${\mathrm{W}}^{\ast }$-algebras initiated by Farah, Hart, and Sherman. By relating the new theory with the original $1$-bounded entropy, we show that if $h(\mathcal{N}:\mathcal{M})\ge 0$, then there exists an embedding of $\mathcal{M}$ into a matrix ultraproduct $\mathcal{Q}={\prod }_{n\to \mathcal{U}}{M}_n(\mathbb{C})$ such that $h(\mathcal{N}:\mathcal{Q})$ is arbitrarily close to $h(\mathcal{N}:\mathcal{M})$. We deduce if all embeddings of $\mathcal{M}$ into $\mathcal{Q}$ are automorphically equivalent, then $\mathcal{M}$ is strongly $1$-bounded and in fact has $h(\mathcal{M})\le 0$.

arXiv comment63 pages, corrections and minor additions, appendix removed