Non-commutative transport of measure

Abstract: Given self-adjoint operators $X_1, \dots, X_d$ and $Y_1, \dots, Y_d$, it is difficult to tell when the von Neumann algebra generated by the $X_j$'s and $Y_j$'s are isomorphic. Viewing the operators as non-commutative random variables, the isomorphism of von Neumann algebras is equivalent to the existence of a non-commutative function that will push forward the non-commutative probability distribution of $X = (X_1,\dots,X_d)$ to that of $Y =(Y_1,\dots,Y_d)$. It was proved by Guionnet, Shlyakhtenko, and Dabrowski that certain nice non-commutative probability distributions known as free Gibbs laws can be transported to the non-commutative Gaussian distribution, and thus the associated von Neumann algebras are all isomorphic. More recently, we have shown that this transport can be done in a lower triangular manner, so that the von Neumann algebra generated by $X_1, \dots, X_k$ is mapped to the von Neumann algebra generated by $Y_1, \dots, Y_k$ for $k = 1, \dots, d$. Furthermore, this transport arises in a natural way as the large-$n$ limit of classical transport of measure for random variables in the space of $d$-tuples $n \times n$ matrices that approximate $(X_1,\dots,X_d)$ as $n \to \infty$.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

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