PIMS Emergent Research Seminar: Jeet Sampat

For a fixed $d \in \mathbb{N}$, the non-commutative (nc) universe $\mathbb{M}^d$ is defined as the graded (disjoint) union of $d$-tuples of $n \times n$ (complex) matrices over $n \in \mathbb{N}$. A function $f : \mathbb{M}^d \to \mathbb{M}^1$ is said to be a nc function if $f(X_1, \dots, X_d) \in \mathbb{C}^{n \times n}$ whenever $(X_1,\dots,X_d) \in (\mathbb{C}^{n \times n})^d$, $f(X_1 \oplus Y_1,\dots,X_d \oplus Y_d) = f(X_1, \dots, X_d) \oplus f(Y_1, \dots, Y_d)$, and $f(S^{-1} X_1 S, \dots, S^{-1} X_d S) = S^{-1} f(X_1, \dots X_d) S$ whenever $S$ is invertible.

A striking consequence of these properties is that a nc function $f$ is continuous (with respect to some suitable topology) if and only if it is locally bounded. In fact, we can also show that a continuous nc function is automatically holomorphic!

In this talk, I will showcase a mix of algebraic and analytic properties of nc functions. In particular, we have a nc homogeneous Nullstellensatz, a nc version of extending holomorphic functions off subvarieties, a nc maximum modulus principle, and determine biholomorphic maps between certain suitable domains $\Omega \subset \mathbb{M}^d$. In my recent joint work with Orr Moshe Shalit, we obtained these results to classify the quotients of algebras of bounded nc functions over such $\Omega$ up to completely isometric isomorphisms.