自由单调传输
Free Monotone Transport
摘要：
通过求解Monge-Ampère方程的自由模拟，我们证明了Brenier单调传输定理的非交换模拟：如果自伴非交换随机变量的$n$元组$Z_{1},...,Z_{n}$满足正则性条件（其共轭变量$ξ_{1},...,ξ_{n}$在$Z_{1},...,Z_{n}$中应该是解析的，并且$ξ_{j}$在某个解析范数中应该接近于$Z_{j}$），则存在半圆形变量$S_{1},...,S_{n}$的$n$元组的可逆非交换函数$F_{j}$，使得$Z_{j}=F_{j}(S_{1},...,S_{n})$。此外，在$F_{j}=\mathscr{D}_{j}g$和$g$是具有正定Hessian的非交换函数的意义上，$F_{j}$可以被选择为单调的。特别地，我们可以推断出$C^{*}(Z_{1},...,Z_{n})\cong C^{*}(S_{1},...,S_{n})$和$W^{*}(Z_{1},...,Z_{n})\cong L(\mathbb{F}(n))$。因此，我们的条件是一个有用的方式来识别时，$n$ -元组的运营商产生一个自由的群因子。作为结果，我们得到的q-变形的自由群因子$Γ_{q}(\mathbb{R}^{n})$是同构的（足够小的$q$，与绑定依赖于$n$）的自由群因子。我们还部分证明了Voiculescu的一个猜想，自由吉布斯状态，这是一个小扰动的一个周期律产生自由群因子。最后，我们证明了对于矩阵上的某种Gibbs测度，入口单调迁移映射可以很好地近似于由自由单调迁移给出的矩阵迁移映射。
英文摘要：
By solving a free analog of the Monge-Ampère equation, we prove a non-commutative analog of Brenier's monotone transport theorem: if an $n$-tuple of self-adjoint non-commutative random variables $Z_{1},...,Z_{n}$ satisfies a regularity condition (its conjugate variables $ξ_{1},...,ξ_{n}$ should be analytic in $Z_{1},...,Z_{n}$ and $ξ_{j}$ should be close to $Z_{j}$ in a certain analytic norm), then there exist invertible non-commutative functions $F_{j}$ of an $n$-tuple of semicircular variables $S_{1},...,S_{n}$, so that $Z_{j}=F_{j}(S_{1},...,S_{n})$. Moreover, $F_{j}$ can be chosen to be monotone, in the sense that $F_{j}=\mathscr{D}_{j}g$ and $g$ is a non-commutative function with a positive definite Hessian. In particular, we can deduce that $C^{*}(Z_{1},...,Z_{n})\cong C^{*}(S_{1},...,S_{n})$ and $W^{*}(Z_{1},...,Z_{n})\cong L(\mathbb{F}(n))$. Thus our condition is a useful way to recognize when an $n$-tuple of operators generate a free group factor. We obtain as a consequence that the q-deformed free group factors $Γ_{q}(\mathbb{R}^{n})$ are isomorphic (for sufficiently small $q$, with bound depending on $n$) to free group factors. We also partially prove a conjecture of Voiculescu by showing that free Gibbs states which are small perturbations of a semicircle law generate free group factors. Lastly, we show that entrywise monotone transport maps for certain Gibbs measure on matrices are well-approximated by the matricial transport maps given by free monotone transport.
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