## Mathematical Physics and Operator Algebras

•  Brent Nelson, Michigan State University
•  Complex analysis applied to operator algebras
•  03/18/2021
•  5:00 PM - 5:50 PM
•  Online (virtual meeting) (Virtual Meeting Link)
•  Brent Nelson (banelson@msu.edu)

Given a positive definite matrix $D\in M_n(\mathbb{C})$ with $\text{Tr}(D)=1$, one can define a linear functional $\varphi\colon M_n(\mathbb{C})\to \mathbb{C}$ by $\varphi(x):=\text{Tr}(Dx)$ which we call a faithful state. This positive definite matrix also encodes a noncommutative dynamical system through $x\mapsto D^{it} x D^{-it}$ for $t\in \mathbb{R}$. From the perspective of operator algebras, it is useful to encode this dynamical system as... well, an algebra of operators. More precisely, as a $*$-algebra $\mathcal{M}$ containing $M_n(\mathbb{C})$ in a way that remembers the action of $\mathbb{R}$. In the general (infinite dimensional) setting, this is accomplished using crossed products and Tomita–Takesaki theory. In this talk, I will apply these methods to the more modest finite dimensional case, and show how a little bit of complex analysis allows one to find the analogue of $\text{Tr}$ on this larger $*$-algebra $\mathcal{M}$. (This talk will assume some familiarity with linear algebra and complex analysis, but nothing further.)

## Contact

Department of Mathematics
Michigan State University