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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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Covering entropy for types in tracial W*-algebras
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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- Award ID(s):
- 2002826
- PAR ID:
- 10432667
- Date Published:
- Journal Name:
- Journal of Logic and Analysis
- Volume:
- 15
- ISSN:
- 1759-9008
- 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/10.4115/jla.2023.15.2
