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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 $$h(\cM) \leq 0$$.
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Duality for Optimal Couplings in Free Probability
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 .
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- PAR ID:
- 10432358
- Date Published:
- Journal Name:
- Communications in Mathematical Physics
- Volume:
- 396
- Issue:
- 3
- ISSN:
- 0010-3616
- Page Range / eLocation ID:
- 903 to 981
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00220-022-04480-0
