Covering entropy for types in tracial W*-algebras

We study embeddings of tracial W^*-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 (X₁(N),X₂(N),) having approximately the same $*$-moments as the generators (X₁,X₂,) of a given tracial 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 W^*-algebras initiated by Farah, Hart, and Sherman. By relating the new theory with the original $1$-bounded entropy, we show that if M is a separable tracial W^*-algebra with h(:) ≥ 0, then there exists an embedding of into a matrix ultraproduct = ∏n → Mₙ() such that h(:) is arbitrarily close to h(:). We deduce that if all embeddings of into are automorphically equivalent, then is strongly $1$-bounded and in fact has h() ≤ 0.