Speaker
Brent Nelson, Michigan State University
Abstract
Voiculescu’s asymptotic freeness theorem from 1991 revealed a deep connection between operator algebras and random matrix theory: d-tuples $(X_1,\ldots, X_d)$ of independent $n\times n$ Gaussian Unitary Ensembles (GUE’s) converge to a d-tuple of self-adjoint operators $(s_1,\ldots, s_d)$ in a tracial von Neumann algebra $(M,\tau)$ in the sense that $\frac{1}{n} Tr(p(X_1,\ldots, X_d)) \to \tau(p(s_1,\ldots, s_d))$ almost surely as $n\to \infty$ for any non-commutative polynomial $p$. In fact, $s_1,\ldots, s_d$ are freely independent semicircular operators and $(M,\tau)$ is $L(\mathbb{F}_n)$ equipped with its canonical trace. In 2005, Haagerup–Thorbjørnsen showed that one further has $\| p(X_1,\ldots, X_d) \| \to \| p(s_1,\ldots, s_d) \|$ almost surely for all non-commutative polynomials $p$. The combination of these convergences is called strong convergence, and it has recently produced many applications to other fields, not to mention much activity within the theories of free probability and random matrices. In joint work with David Jekel, Yoonkyeong Lee, and Jennifer Pi, we establish a framework for strong convergence in the operator-valued setting, where the tracial state $\tau\colon M\to \mathbb{C}$ is replaced by a conditional expectation $E\colon M\to B$ onto a von Neumann subalgebra. In this context, any completely positive map $\eta\colon B\to M_d(B)$ determines a natural generalization of $(s_1,\ldots, s_d)$ called a $(B,\eta)$-semicircular family, and similarly any completely positive map $\eta\colon M_n(\mathbb{C})\to M_{dn}(\mathbb{C})$ determines a generalization of $(X_1,\ldots, X_d)$ called an $\eta$-Gaussian family. We show that when $\eta$ on $B$ is suitably approximated by a sequence of finite dimensional maps $\eta^{(n)}$ on $M_n(\mathbb{C})$, then one has strong convergence of the $\eta^{(n)}$-Gaussian families to the $(B,\eta)$-semicircular family.
Brent Nelson, Michigan State University
Abstract
Voiculescu’s asymptotic freeness theorem from 1991 revealed a deep connection between operator algebras and random matrix theory: d-tuples $(X_1,\ldots, X_d)$ of independent $n\times n$ Gaussian Unitary Ensembles (GUE’s) converge to a d-tuple of self-adjoint operators $(s_1,\ldots, s_d)$ in a tracial von Neumann algebra $(M,\tau)$ in the sense that $\frac{1}{n} Tr(p(X_1,\ldots, X_d)) \to \tau(p(s_1,\ldots, s_d))$ almost surely as $n\to \infty$ for any non-commutative polynomial $p$. In fact, $s_1,\ldots, s_d$ are freely independent semicircular operators and $(M,\tau)$ is $L(\mathbb{F}_n)$ equipped with its canonical trace. In 2005, Haagerup–Thorbjørnsen showed that one further has $\| p(X_1,\ldots, X_d) \| \to \| p(s_1,\ldots, s_d) \|$ almost surely for all non-commutative polynomials $p$. The combination of these convergences is called strong convergence, and it has recently produced many applications to other fields, not to mention much activity within the theories of free probability and random matrices. In joint work with David Jekel, Yoonkyeong Lee, and Jennifer Pi, we establish a framework for strong convergence in the operator-valued setting, where the tracial state $\tau\colon M\to \mathbb{C}$ is replaced by a conditional expectation $E\colon M\to B$ onto a von Neumann subalgebra. In this context, any completely positive map $\eta\colon B\to M_d(B)$ determines a natural generalization of $(s_1,\ldots, s_d)$ called a $(B,\eta)$-semicircular family, and similarly any completely positive map $\eta\colon M_n(\mathbb{C})\to M_{dn}(\mathbb{C})$ determines a generalization of $(X_1,\ldots, X_d)$ called an $\eta$-Gaussian family. We show that when $\eta$ on $B$ is suitably approximated by a sequence of finite dimensional maps $\eta^{(n)}$ on $M_n(\mathbb{C})$, then one has strong convergence of the $\eta^{(n)}$-Gaussian families to the $(B,\eta)$-semicircular family.