 Main
Evolution equations in noncommutative probability
 Jekel, David Andrew
 Advisor(s): Shlyakhtenko, Dimitri
Abstract
The thesis presents two applications of evolution equations for noncommutative variables to the theory of noncommutative probability and von Neumann algebras.
In the first part, noncommutative processes $(X_t)_{t \in [0,T]}$ with boolean, free, monotone, or antimonotone independent increments, under certain continuity and boundedness assumptions, are classified in terms of certain evolution equations for their $F$transforms $F_{X_t}(z) = (E[(z  X_t)^{1}])^{1}$. This classification is done in the setting of operatorvalued noncommutative probability, in which the expectation takes values in a $\mathrm{C}^*$algebra $\mathcal{B}$ rather than $\mathbb{C}$. Thus, the $F$transform is a function of an operator variable $z$ from (matrices over) $\mathcal{B}$, and it is understood through the theory of fully matricial or noncommutative functions, an operatorvalued analogue of complex analysis. The classification of these processes generalizes previous work on the L {e}vyHin\v{c}in formula for processes with independent and stationary increments, and it leads to BercoviciPatatype bijections between the processes with independent increments for the four different types of independence. We also describe a canonical model for each process with independent increments using operators on a Fock space. In fact, the interaction between operator models and analytic function theory is a major theme of the first part, and leads to a new ``coupling'' technique to prove estimates for the noncommutative central limit theorem and for Loewner chains.
In the second part, we strengthen the probabilistic, informationtheoretic, and transporttheoretic connections between asymptotic random matrix theory and tracial $\mathrm{W}^*$algebras through the study of functions and differential equations for several noncommuting selfadjoint variables. We consider a random variable $X^{(n)}$ in $M_n(\mathbb{C})_{\text{sa}}^d$ given by a probability distribution
\[
d\mu^{(n)}(x) = \frac{1}{\int e^{n^2 V^{(n)}}} e^{n^2 V^{(n)}(x)}\,dx,
\]
where $V^{(n)}: M_n(\C)_{sa}^d \to \R$ is uniformly convex and semiconcave. We assume that $(\nabla V^{(n)})_{n \in \N}$ is asymptotically approximable by trace polynomials, which means that $\nabla V^{(n)}$ behaves asymptotically like some element $f$ from a certain space of ``functions of $d$ selfadjoint variables from a tracial $\mathrm{W}^*$algebra.''
Then we show first that $X^{(n)}$ almost surely converges in noncommutative law to some $d$tuple $X$ from a tracial $\mathrm{W}^*$algebra $(\mathcal{M},\tau)$, meaning that $(1/n) \Tr(p(X^{(n)})) \to \tau(p(X))$ almost surely for every noncommutative polynomial $p$ (which is comparable to earlier known results). The strategy to prove convergence of the expectation $E[(1/n) \Tr(p(X^{(n)}))]$ is to show that the heat semigroup associated to the measure $\mu^{(n)}$ preserves asymptotic approximability by trace polynomials. The same method leads to a new conditional version of this result, which shows that if $k < d$ and if $(f^{(n)})$ is asymptotically approximable by polynomials, then so is the function $g^{(n)}$ given by $g^{(n)}(X_1^{(n)},\dots,X_k^{(n)}) = E[f^{(n)}(X^{(n)})  X_1^{(n)}, \dots, X_k^{(n)}]$. Understanding the large$n$ behavior of such conditional expectations is a key step in showing our second main result that the classical entropy of $X^{(n)}$, after renormalization, converges to Voiculescu's nonmicrostates free entropy $\chi^*(X)$ (and an analogous result for conditional entropy given $X_1^{(n)}$, \dots, $X_k^{(n)}$). In particular, we obtain a new proof of the result from a 2017 preprint of Dabrowski that $\chi^*(X)$ agrees with the microstates free entropy $\chi(X)$ for any $X$ that arises from such random matrix models.
The final main result studies the large$n$ behavior of certain functions $F^{(n)}$ that transport the measure $\mu^{(n)}$ to the distribution $\sigma_1^{(n)}$ of a standard Gaussian selfadjoint $d$tuple $Z^{(n)}$. The transport map $F^{(n)}$ is obtained by the same construction as in Otto and Villani's famous proof of the Talagrand inequality based on heat semigroups and transport equations. Using successive conditioning, we can obtain a transport function $F^{(n)}$ that is lowertriangular in the sense that
\[
F^{(n)}(x_1,\dots,x_d) = (F_1^{(n)}(x_1), F_2^{(n)}(x_1,x_2),\dots,F_d^{(n)}(x_1,\dots,x_d)),
\]
where $x = (x_1,\dots,x_d) \in M_n(\C)^d$. We show that $F^{(n)}$ is asymptotically approximable by trace polynomials as $n \to \infty$, and consequently, in the large$n$ limit, we obtain an isomorphism $\mathrm{W}^*(X_1,\dots,X_d) \to \mathrm{W}^*(Z_1,\dots,Z_d)$ that maps $\mathrm{W}^*(X_1,\dots,X_k)$ to $\mathrm{W}^*(Z_1,\dots,Z_k)$ for every $k = 1, \dots, d$. As an application, we show that this statement holds when $X$ itself is given by $X_j = Z_j + \delta p_j(Z)$ where $Z$ is a free semicircular $d$tuple, $p_j$ is a selfadjoint noncommutative polynomial, and $\delta$ is sufficiently small, depending on $p_1$, \dots, $p_d$.
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