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

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