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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 $$h(\cM) \leq 0$$. 

Abstract We study the free probabilistic analog of optimal couplings for the quadratic cost, where classical probability spaces are replaced by tracial von Neumann algebras, and probability measures on $${\mathbb {R}}^m$$ R m are replaced by non-commutative laws of m -tuples. We prove an analog of the Monge–Kantorovich duality which characterizes optimal couplings of non-commutative laws with respect to Biane and Voiculescu’s non-commutative $$L^2$$ L 2 -Wasserstein distance using a new type of convex functions. As a consequence, we show that if ( X ,  Y ) is a pair of optimally coupled m -tuples of non-commutative random variables in a tracial $$\mathrm {W}^*$$ W ∗ -algebra $$\mathcal {A}$$ A , then $$\mathrm {W}^*((1 - t)X + tY) = \mathrm {W}^*(X,Y)$$ W ∗ ( ( 1 - t ) X + t Y ) = W ∗ ( X , Y ) for all $$t \in (0,1)$$ t ∈ ( 0 , 1 ) . Finally, we illustrate the subtleties of non-commutative optimal couplings through connections with results in quantum information theory and operator algebras. For instance, two non-commutative laws that can be realized in finite-dimensional algebras may still require an infinite-dimensional algebra to optimally couple. Moreover, the space of non-commutative laws of m -tuples is not separable with respect to the Wasserstein distance for $$m > 1$$ m > 1 . 

We develop the complex-analytic viewpoint on the tree convolutions studied by the second author and Weihua Liu in Jekel and Liu (2020), which generalize the free, boolean, monotone, and orthogonal convolutions. In particular, for each rooted subtree T of the N-regular tree (with vertices labeled by alternating strings), we define the convolution \boxplus_T (µ1, . . . , µN) for arbitrary probability measures µ1, . . . , µN on R using a certain fixed-point equation for the Cauchy transforms. The convolution operations respect the operad structure of the tree operad from Jekel and Liu (2020). We prove a general limit theorem for iterated T -free convolution similar to Bercovici and Pata’s results in the free case (Bercovici and Pata 1999), and we deduce limit theorems for measures in the domain of attraction of each of the classical stable laws. 

Abstract This paper gives a free entropy theoretic perspective on amenable absorption results for free products of tracial von Neumann algebras. In particular, we give the 1st free entropy proof of Popa’s famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer’s results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using $$1$$-bounded entropy. We show that if $${\mathcal{M}} = {\mathcal{P}} * {\mathcal{Q}}$$, then $${\mathcal{P}}$$ absorbs any subalgebra of $${\mathcal{M}}$$ that intersects it diffusely and that has $$1$$-bounded entropy zero (which includes amenable and property Gamma algebras as well as many others). In fact, for a subalgebra $${\mathcal{P}} \leq{\mathcal{M}}$$ to have this absorption property, it suffices for $${\mathcal{M}}$$ to admit random matrix models that have exponential concentration of measure and that “simulate” the conditional expectation onto $${\mathcal{P}}$$.